A Falling Stone Trades Height for Speed at Exactly the Same Rate Throughout
Derive the expressions for potential and kinetic energy, tell translational from rotational and vibrational kinetic energy, follow energy through its chemical, electrical, nuclear and other forms, and prove that the total stays constant for a freely falling body.
Where does the energy go when a stone falls?
Hold a stone above the ground and it has energy stored in it — energy you had to supply by lifting it. Let it go and that stored energy does not vanish. It turns into speed.
And it does so with a precision that is easy to check. Drop a stone from m and at every instant during the fall:
- The energy it has lost by descending is exactly times the drop
- The energy it has gained as motion is exactly
- Those two amounts are equal, all the way down
So the total never changes. At the top the stone has all its energy as height and none as speed; halfway down it has half of each; at the ground it has all speed and no height. The stone is trading one form of energy for the other at a fixed exchange rate, and nothing is lost in the transaction.
That statement is the principle of conservation of energy, and this part of the chapter builds up to proving it for the simplest case.
To get there you need three things.
- Expressions for the two mechanical energies — potential energy from height, kinetic energy from speed — derived rather than quoted
- A clear idea of what kinetic energy includes, because a body can be moving in three quite different ways at once
- A survey of the other forms energy takes — chemical, electrical, heat, nuclear, sound, light — and how one becomes another
And then the principle itself, which is one of the very few statements in physics with no known exception. Energy can be converted from any form into any other, but the total amount in an isolated system stays exactly the same. It can neither be created nor destroyed.
Unless a question says otherwise, **take m s** for the numericals here.
This page covers the third part of the ICSE Class 10 Physics chapter on force, work, power and energy: potential and kinetic energy, the forms of energy, and the conservation of energy.
And it does so with a precision that is easy to check. Drop a stone from m and at every instant during the fall:
- The energy it has lost by descending is exactly times the drop
- The energy it has gained as motion is exactly
- Those two amounts are equal, all the way down
So the total never changes. At the top the stone has all its energy as height and none as speed; halfway down it has half of each; at the ground it has all speed and no height. The stone is trading one form of energy for the other at a fixed exchange rate, and nothing is lost in the transaction.
That statement is the principle of conservation of energy, and this part of the chapter builds up to proving it for the simplest case.
To get there you need three things.
- Expressions for the two mechanical energies — potential energy from height, kinetic energy from speed — derived rather than quoted
- A clear idea of what kinetic energy includes, because a body can be moving in three quite different ways at once
- A survey of the other forms energy takes — chemical, electrical, heat, nuclear, sound, light — and how one becomes another
And then the principle itself, which is one of the very few statements in physics with no known exception. Energy can be converted from any form into any other, but the total amount in an isolated system stays exactly the same. It can neither be created nor destroyed.
Unless a question says otherwise, **take m s** for the numericals here.
This page covers the third part of the ICSE Class 10 Physics chapter on force, work, power and energy: potential and kinetic energy, the forms of energy, and the conservation of energy.
Formula
How are the formulas for potential and kinetic energy derived?
Both come from the definition of work. Potential energy is the work done in lifting the body; kinetic energy is the work done in getting it moving.
**The derivation of .** To raise a body of mass slowly through a height , you must apply an upward force just equal to its weight , and the displacement is along that force. So
And that work is recoverable. Release the body and gravity does exactly of work on the way down, which is what makes this energy "potential" — stored, and available again.
**The derivation of .** Let a constant force act on a body of mass at rest and bring it to a speed over a distance . From the equations of motion with ,
The work done is , and , so
The acceleration cancelled, which is the important step — it means the kinetic energy depends only on the mass and the final speed, not on how hard or how gently the body was accelerated.
Worked example 1 — potential energy. Find the potential energy of a kg body held m above the ground.
Worked example 2 — kinetic energy. Find the kinetic energy of a kg body moving at m s.
Worked example 3 — the effect of doubling the speed. What happens to the kinetic energy if the speed of the same body is doubled to m s?
Four times as much, not twice. The speed appears squared, so doubling it multiplies the kinetic energy by four and tripling it by nine. That is why stopping distances grow so steeply with speed — the brakes have four times as much energy to remove.
Worked example 4 — finding the speed from the energy. A body of mass kg has a kinetic energy of J. Find its speed and its momentum.
Check with the energy-momentum relation. Since and , it follows that
The same value, and that relation is worth knowing because some questions give the momentum rather than the speed.
Worked example 5 — height from kinetic energy. A ball of mass kg is thrown up with J of kinetic energy. How high does it rise?
All of the kinetic energy becomes potential energy at the top:
Two points about potential energy that examiners test. It is measured relative to a chosen reference level, usually the ground — so a body on a table has potential energy with respect to the floor but none with respect to the table top. And it depends only on the height gained, not on the path taken, which is why a straight lift and a long ramp to the same height store the same energy.
**The derivation of .** To raise a body of mass slowly through a height , you must apply an upward force just equal to its weight , and the displacement is along that force. So
And that work is recoverable. Release the body and gravity does exactly of work on the way down, which is what makes this energy "potential" — stored, and available again.
**The derivation of .** Let a constant force act on a body of mass at rest and bring it to a speed over a distance . From the equations of motion with ,
The work done is , and , so
The acceleration cancelled, which is the important step — it means the kinetic energy depends only on the mass and the final speed, not on how hard or how gently the body was accelerated.
Worked example 1 — potential energy. Find the potential energy of a kg body held m above the ground.
Worked example 2 — kinetic energy. Find the kinetic energy of a kg body moving at m s.
Worked example 3 — the effect of doubling the speed. What happens to the kinetic energy if the speed of the same body is doubled to m s?
Four times as much, not twice. The speed appears squared, so doubling it multiplies the kinetic energy by four and tripling it by nine. That is why stopping distances grow so steeply with speed — the brakes have four times as much energy to remove.
Worked example 4 — finding the speed from the energy. A body of mass kg has a kinetic energy of J. Find its speed and its momentum.
Check with the energy-momentum relation. Since and , it follows that
The same value, and that relation is worth knowing because some questions give the momentum rather than the speed.
Worked example 5 — height from kinetic energy. A ball of mass kg is thrown up with J of kinetic energy. How high does it rise?
All of the kinetic energy becomes potential energy at the top:
Two points about potential energy that examiners test. It is measured relative to a chosen reference level, usually the ground — so a body on a table has potential energy with respect to the floor but none with respect to the table top. And it depends only on the height gained, not on the path taken, which is why a straight lift and a long ramp to the same height store the same energy.
What are the three forms of kinetic energy?
Translational, rotational and vibrational — depending on whether the body moves from place to place, turns about an axis, or moves to and fro about a fixed position.
Translational kinetic energy belongs to a body whose whole mass moves from one place to another along a path.
- A car travelling on a road
- A bullet fired from a gun
- A stone falling freely
- Water flowing in a river
**This is the only one the formula is used for at this level, and every numerical in the chapter assumes translational motion.
Rotational kinetic energy belongs to a body turning about an axis, where the body as a whole may not move anywhere at all.
- A spinning top, which stays in one place while turning rapidly
- A rotating fan blade or a ceiling fan
- A merry-go-round in a park
- The earth spinning on its own axis
- The wheels of a moving cycle, which have both kinds at once
That last example is the one to think about. A rolling wheel is moving forward and turning, so it has translational kinetic energy because of its forward motion and rotational kinetic energy because of its spin. The two add, which is why a rolling ball is harder to stop than a sliding one of the same mass and speed.
Vibrational kinetic energy belongs to a body or its particles moving to and fro about a mean position.
- The prongs of a struck tuning fork
- A plucked guitar string
- The atoms of a solid, which vibrate about fixed positions in the lattice
- The molecules of a gas, which have vibrational energy in addition to their translational motion
And here is the link that makes this classification worth learning. The vibrational kinetic energy of the particles of a body is its heat energy. Heating a substance means making its particles vibrate faster, and temperature is a measure of the average kinetic energy of those particles. So the three forms of kinetic energy are not a tidy list — the third one is the bridge between mechanics and heat.
Worked example — a rolling body.** A wheel of mass kg rolls along the ground at m s. Find its translational kinetic energy.
Its total kinetic energy is more than this, because the wheel is also spinning, and the rotational part must be added. Calculating that part needs the moment of inertia, which belongs to Class 11 — so at this level a question about a rolling body asks only for the translational part, and it should say so.
One boundary case worth noticing. A spinning top standing on the same spot has kinetic energy even though its centre is not moving anywhere. So "kinetic energy" does not require the body to go anywhere — it requires its particles to be moving, and in a spinning top every particle except those on the axis is moving in a circle.
Translational kinetic energy belongs to a body whose whole mass moves from one place to another along a path.
- A car travelling on a road
- A bullet fired from a gun
- A stone falling freely
- Water flowing in a river
**This is the only one the formula is used for at this level, and every numerical in the chapter assumes translational motion.
Rotational kinetic energy belongs to a body turning about an axis, where the body as a whole may not move anywhere at all.
- A spinning top, which stays in one place while turning rapidly
- A rotating fan blade or a ceiling fan
- A merry-go-round in a park
- The earth spinning on its own axis
- The wheels of a moving cycle, which have both kinds at once
That last example is the one to think about. A rolling wheel is moving forward and turning, so it has translational kinetic energy because of its forward motion and rotational kinetic energy because of its spin. The two add, which is why a rolling ball is harder to stop than a sliding one of the same mass and speed.
Vibrational kinetic energy belongs to a body or its particles moving to and fro about a mean position.
- The prongs of a struck tuning fork
- A plucked guitar string
- The atoms of a solid, which vibrate about fixed positions in the lattice
- The molecules of a gas, which have vibrational energy in addition to their translational motion
And here is the link that makes this classification worth learning. The vibrational kinetic energy of the particles of a body is its heat energy. Heating a substance means making its particles vibrate faster, and temperature is a measure of the average kinetic energy of those particles. So the three forms of kinetic energy are not a tidy list — the third one is the bridge between mechanics and heat.
Worked example — a rolling body.** A wheel of mass kg rolls along the ground at m s. Find its translational kinetic energy.
Its total kinetic energy is more than this, because the wheel is also spinning, and the rotational part must be added. Calculating that part needs the moment of inertia, which belongs to Class 11 — so at this level a question about a rolling body asks only for the translational part, and it should say so.
One boundary case worth noticing. A spinning top standing on the same spot has kinetic energy even though its centre is not moving anywhere. So "kinetic energy" does not require the body to go anywhere — it requires its particles to be moving, and in a spinning top every particle except those on the axis is moving in a circle.
What forms does energy take, and how does one form become another?
Energy exists in several forms, and every device you use is a converter from one form to another.
The forms named in the syllabus, each with what it is:
- Mechanical energy — the potential and kinetic energy of a body, as in a stretched spring, flowing water or a moving vehicle
- Chemical energy — the energy stored in the bonds of substances, as in food, fuels, dry cells and explosives
- Heat energy — the vibrational kinetic energy of the particles of a body, as in steam or a hot iron
- Electrical energy — the energy carried by charges moving through a circuit
- Nuclear energy — the energy stored in the nucleus, released in fission and fusion
- Sound energy — the energy carried by the compressions and rarefactions of a wave through a medium
- Light energy — the energy carried by electromagnetic radiation in the visible range
And the conversions, which is what questions actually ask for.
- Chemical to heat and light — burning coal, wood or gas in a stove; a candle
- Chemical to electrical — a dry cell or a car battery in use
- Electrical to chemical — charging a battery, and electrolysis
- Electrical to heat — an electric iron, a heater, an immersion rod
- Electrical to light — a bulb or a light-emitting diode
- Electrical to sound — a loudspeaker or a doorbell
- Electrical to mechanical — a fan, a mixer or an electric motor
- Mechanical to electrical — a dynamo on a cycle, a generator at a power station
- Light to electrical — a solar cell
- Sound to electrical — a microphone
- Heat to mechanical — a steam engine or a steam turbine
- Nuclear to heat and then to electrical — a nuclear power station
- Mechanical to heat — rubbing your palms together, or a drill bit warming as it cuts
Worked example — following the chain in a power station. Trace the energy conversions in a thermal power station that burns coal to light a bulb in a house.
- Chemical energy of the coal becomes heat on burning
- Heat turns water into steam, giving the steam mechanical energy
- Mechanical energy of the steam turns a turbine and a generator, producing electrical energy
- Electrical energy travels to the house and becomes light and heat in the bulb
Four conversions in one chain, and at each step some energy is lost as unwanted heat — which is why no power station converts all of a fuel's chemical energy into electricity.
The honest point to make about "energy loss". When we say energy is lost, we never mean it has been destroyed. It has been converted into a form that is not useful to us, almost always low-grade heat spread thinly through the surroundings. The total is unchanged; only its usefulness has fallen — and that distinction is the difference between a correct answer and a marked error in this chapter.
A note on the word "renewable". Solar, wind, tidal and hydro sources are called renewable because they are replenished continuously; coal, petroleum and natural gas are non-renewable because their stocks took an immense time to form and are being used far faster than they are replaced. Conservation of energy says nothing about this — the physical law is about the total amount, while the practical problem is about which forms are available to us.
The forms named in the syllabus, each with what it is:
- Mechanical energy — the potential and kinetic energy of a body, as in a stretched spring, flowing water or a moving vehicle
- Chemical energy — the energy stored in the bonds of substances, as in food, fuels, dry cells and explosives
- Heat energy — the vibrational kinetic energy of the particles of a body, as in steam or a hot iron
- Electrical energy — the energy carried by charges moving through a circuit
- Nuclear energy — the energy stored in the nucleus, released in fission and fusion
- Sound energy — the energy carried by the compressions and rarefactions of a wave through a medium
- Light energy — the energy carried by electromagnetic radiation in the visible range
And the conversions, which is what questions actually ask for.
- Chemical to heat and light — burning coal, wood or gas in a stove; a candle
- Chemical to electrical — a dry cell or a car battery in use
- Electrical to chemical — charging a battery, and electrolysis
- Electrical to heat — an electric iron, a heater, an immersion rod
- Electrical to light — a bulb or a light-emitting diode
- Electrical to sound — a loudspeaker or a doorbell
- Electrical to mechanical — a fan, a mixer or an electric motor
- Mechanical to electrical — a dynamo on a cycle, a generator at a power station
- Light to electrical — a solar cell
- Sound to electrical — a microphone
- Heat to mechanical — a steam engine or a steam turbine
- Nuclear to heat and then to electrical — a nuclear power station
- Mechanical to heat — rubbing your palms together, or a drill bit warming as it cuts
Worked example — following the chain in a power station. Trace the energy conversions in a thermal power station that burns coal to light a bulb in a house.
- Chemical energy of the coal becomes heat on burning
- Heat turns water into steam, giving the steam mechanical energy
- Mechanical energy of the steam turns a turbine and a generator, producing electrical energy
- Electrical energy travels to the house and becomes light and heat in the bulb
Four conversions in one chain, and at each step some energy is lost as unwanted heat — which is why no power station converts all of a fuel's chemical energy into electricity.
The honest point to make about "energy loss". When we say energy is lost, we never mean it has been destroyed. It has been converted into a form that is not useful to us, almost always low-grade heat spread thinly through the surroundings. The total is unchanged; only its usefulness has fallen — and that distinction is the difference between a correct answer and a marked error in this chapter.
A note on the word "renewable". Solar, wind, tidal and hydro sources are called renewable because they are replenished continuously; coal, petroleum and natural gas are non-renewable because their stocks took an immense time to form and are being used far faster than they are replaced. Conservation of energy says nothing about this — the physical law is about the total amount, while the practical problem is about which forms are available to us.
How do you prove that the total energy of a falling body stays constant?
Compute the potential and kinetic energies separately at any height during the fall, add them, and show that the height cancels out.
The general proof. Let a body of mass be dropped from rest at a height above the ground, and consider it when it has fallen to a height above the ground.
Its potential energy at that point, measured from the ground, is
Its speed there follows from with , and a fall of :
so its kinetic energy is
Adding the two:
**The cancels completely.** So the total mechanical energy is at every point of the fall, whatever the value of — which is exactly the principle of conservation of energy for this case.
Worked example — the same proof in numbers. A ball of mass kg is dropped from a height of m. Find its potential and kinetic energies at the top, at a height of m, and at the ground.
At the top, m:
**At a height of m**, the ball has fallen m, so :
At the ground, , and the ball has fallen the full m, so and m s:
**The total is J at all three points, and at the halfway height the energy is split exactly in half. That last fact is a useful check: the potential and kinetic energies of a freely falling body are equal at half the original height.
The simple pendulum, applied qualitatively. A pendulum bob swings between two extreme positions through a lowest point, and its energy moves back and forth between the two forms:
- At either extreme position the bob is momentarily at rest and at its highest point, so its kinetic energy is zero and its potential energy is maximum
- At the lowest point the bob is moving fastest and is at its lowest, so its potential energy is zero and its kinetic energy is maximum
- At every point in between the sum of the two is the same
Worked example — a pendulum numerically.** A pendulum bob of mass kg is drawn aside so that it rises m above its lowest point, and released. Find its speed at the lowest point.
All the potential energy becomes kinetic energy:
The mass cancelled again, so a heavy bob and a light one released from the same height arrive at the lowest point with the same speed.
Why a real pendulum eventually stops, and this is the boundary case. The proof assumed no air resistance and no friction at the support. In practice both are present, so at each swing a little of the mechanical energy is converted into heat and sound and the amplitude decreases. The total energy is still conserved — it has simply gone into the air and the support rather than staying in the bob. A pendulum in a vacuum with a frictionless pivot would swing forever, and saying that is what distinguishes an understanding of the principle from a memory of it.
The general proof. Let a body of mass be dropped from rest at a height above the ground, and consider it when it has fallen to a height above the ground.
Its potential energy at that point, measured from the ground, is
Its speed there follows from with , and a fall of :
so its kinetic energy is
Adding the two:
**The cancels completely.** So the total mechanical energy is at every point of the fall, whatever the value of — which is exactly the principle of conservation of energy for this case.
Worked example — the same proof in numbers. A ball of mass kg is dropped from a height of m. Find its potential and kinetic energies at the top, at a height of m, and at the ground.
At the top, m:
**At a height of m**, the ball has fallen m, so :
At the ground, , and the ball has fallen the full m, so and m s:
**The total is J at all three points, and at the halfway height the energy is split exactly in half. That last fact is a useful check: the potential and kinetic energies of a freely falling body are equal at half the original height.
The simple pendulum, applied qualitatively. A pendulum bob swings between two extreme positions through a lowest point, and its energy moves back and forth between the two forms:
- At either extreme position the bob is momentarily at rest and at its highest point, so its kinetic energy is zero and its potential energy is maximum
- At the lowest point the bob is moving fastest and is at its lowest, so its potential energy is zero and its kinetic energy is maximum
- At every point in between the sum of the two is the same
Worked example — a pendulum numerically.** A pendulum bob of mass kg is drawn aside so that it rises m above its lowest point, and released. Find its speed at the lowest point.
All the potential energy becomes kinetic energy:
The mass cancelled again, so a heavy bob and a light one released from the same height arrive at the lowest point with the same speed.
Why a real pendulum eventually stops, and this is the boundary case. The proof assumed no air resistance and no friction at the support. In practice both are present, so at each swing a little of the mechanical energy is converted into heat and sound and the amplitude decreases. The total energy is still conserved — it has simply gone into the air and the support rather than staying in the bob. A pendulum in a vacuum with a frictionless pivot would swing forever, and saying that is what distinguishes an understanding of the principle from a memory of it.
Exam tip
Which habits protect the marks in an energy question?
State the reference level for potential energy, write both energies separately, and add them only at the end. Questions in this chapter are short, and the marks sit in the setting-out.
- Say what the height is measured from — usually the ground — because potential energy has no meaning without a reference level
- Square the speed, not the mass, in ; doubling the speed quadruples the energy
- Use the vertical height only in ; the path does not matter
- **Take m s unless told otherwise, and state the value you used
- For a free fall, get the speed from , which is the distance fallen and not the height remaining
- Check that comes out the same at every point — that is the whole content of the conservation proof
- Use the half-height check: potential and kinetic energies are equal at half the original height
- For a pendulum, name the extreme and lowest positions and say which energy is maximum at each
- Give the unit — joule for both energies, and metre per second for speeds
- Say "converted into heat and sound" rather than "lost", when explaining why a real pendulum stops
The misconception to name. Energy is never used up or destroyed. When a torch battery goes flat, its chemical energy has been converted into light and heat, not consumed — and when a pendulum stops, its mechanical energy has gone into warming the air rather than disappearing. A question asking "where has the energy gone?" always has an answer, and "it was lost" is not it.
A second trap. Confusing the distance fallen with the height above the ground in the free-fall calculation. At a height of m in a m fall, the body has fallen m, so and not .** Using the wrong one gives a total that does not come out constant — which is itself the signal that the two have been swapped.
- Say what the height is measured from — usually the ground — because potential energy has no meaning without a reference level
- Square the speed, not the mass, in ; doubling the speed quadruples the energy
- Use the vertical height only in ; the path does not matter
- **Take m s unless told otherwise, and state the value you used
- For a free fall, get the speed from , which is the distance fallen and not the height remaining
- Check that comes out the same at every point — that is the whole content of the conservation proof
- Use the half-height check: potential and kinetic energies are equal at half the original height
- For a pendulum, name the extreme and lowest positions and say which energy is maximum at each
- Give the unit — joule for both energies, and metre per second for speeds
- Say "converted into heat and sound" rather than "lost", when explaining why a real pendulum stops
The misconception to name. Energy is never used up or destroyed. When a torch battery goes flat, its chemical energy has been converted into light and heat, not consumed — and when a pendulum stops, its mechanical energy has gone into warming the air rather than disappearing. A question asking "where has the energy gone?" always has an answer, and "it was lost" is not it.
A second trap. Confusing the distance fallen with the height above the ground in the free-fall calculation. At a height of m in a m fall, the body has fallen m, so and not .** Using the wrong one gives a total that does not come out constant — which is itself the signal that the two have been swapped.
Did you know
Why do heavy and light stones hit the ground at the same speed?
Drop a one-kilogram stone and a ten-kilogram stone from the same height and, ignoring air resistance, they land at exactly the same speed. The heavier one has ten times the energy, but it also needs ten times as much energy to reach any given speed, and the two factors of ten cancel.
You can see the cancellation in one line. Equating the potential energy at the top to the kinetic energy at the bottom:
The mass has gone. The landing speed depends only on the height and on , so every object dropped from m arrives at m s, whether it is a marble or a boulder.
And the same cancellation ran through three of the worked examples in this page. The stone thrown upward, the pendulum bob and the falling ball all had their masses cancel — because gravity's pull and a body's resistance to being accelerated are both proportional to the same mass. That is a genuinely deep fact about gravity and not an arithmetic accident.
Which raises the obvious objection: a feather clearly does not fall like a stone. The reason is air resistance, not gravity. The feather has a large surface area for its weight, so the air pushes back on it almost as hard as gravity pulls, and it drifts down slowly. Remove the air and the feather and the stone fall together — which is why the argument always carries the phrase "ignoring air resistance".
The same reasoning explains terminal velocity. As a raindrop speeds up, the air resistance on it grows until it exactly balances the weight. After that the net force is zero, the drop stops accelerating, and it falls at a constant speed. From then on the gravitational potential energy it loses is no longer becoming kinetic energy at all — it is going straight into warming the air. The drop's energy is conserved, but its mechanical energy is not.
One more consequence, this time about safety. Because kinetic energy goes as the square of the speed, a vehicle at twice the speed carries four times the energy and needs four times the stopping distance. A vehicle at three times the speed carries nine times as much. That is the physical reason speed limits fall so steeply near schools and markets — not because the speed doubles the danger, but because it quadruples the energy that has to be got rid of.
And the pendulum has a last surprise. Since the mass cancels, the time a pendulum takes to swing does not depend on the mass of its bob either — a heavy bob and a light one on strings of the same length keep the same time. Class 11 shows that the period depends only on the length and on , and the cancellation you have just seen is the first hint of it.
You can see the cancellation in one line. Equating the potential energy at the top to the kinetic energy at the bottom:
The mass has gone. The landing speed depends only on the height and on , so every object dropped from m arrives at m s, whether it is a marble or a boulder.
And the same cancellation ran through three of the worked examples in this page. The stone thrown upward, the pendulum bob and the falling ball all had their masses cancel — because gravity's pull and a body's resistance to being accelerated are both proportional to the same mass. That is a genuinely deep fact about gravity and not an arithmetic accident.
Which raises the obvious objection: a feather clearly does not fall like a stone. The reason is air resistance, not gravity. The feather has a large surface area for its weight, so the air pushes back on it almost as hard as gravity pulls, and it drifts down slowly. Remove the air and the feather and the stone fall together — which is why the argument always carries the phrase "ignoring air resistance".
The same reasoning explains terminal velocity. As a raindrop speeds up, the air resistance on it grows until it exactly balances the weight. After that the net force is zero, the drop stops accelerating, and it falls at a constant speed. From then on the gravitational potential energy it loses is no longer becoming kinetic energy at all — it is going straight into warming the air. The drop's energy is conserved, but its mechanical energy is not.
One more consequence, this time about safety. Because kinetic energy goes as the square of the speed, a vehicle at twice the speed carries four times the energy and needs four times the stopping distance. A vehicle at three times the speed carries nine times as much. That is the physical reason speed limits fall so steeply near schools and markets — not because the speed doubles the danger, but because it quadruples the energy that has to be got rid of.
And the pendulum has a last surprise. Since the mass cancels, the time a pendulum takes to swing does not depend on the mass of its bob either — a heavy bob and a light one on strings of the same length keep the same time. Class 11 shows that the period depends only on the length and on , and the cancellation you have just seen is the first hint of it.
Exam relevance
How does conservation of energy carry into JEE and NEET?
This is foundation work for Class 11 Work, Energy and Power and Systems of Particles and Rotational Motion, and conservation of energy is one of the most reused ideas in JEE Main, JEE Advanced and NEET Physics.
**Where leads. Class 11 generalises it to any conservative force**, defining potential energy through , and then applies it to the spring, where . The fact you meet here — that gravitational potential energy depends on the height and not the path — is precisely the definition of a conservative force, and JEE questions turn on recognising when that is available.
**Where leads.** The relation that you verify here becomes essential in Collisions, where momentum and energy are used together, and again in Class 12 Dual Nature of Radiation for the de Broglie wavelength of a particle of given kinetic energy. NEET uses the momentum-energy relation directly.
Where the free-fall proof leads. It becomes the standard method for any problem involving heights and speeds without times: a block sliding down a curved track, a bead on a wire, a body on a loop. **JEE Advanced sets problems on vertical circular motion that are solved by writing at two points and equating them, which is the proof of this section applied twice.
Where the three forms of kinetic energy lead.** Class 11 quantifies the rotational part as , and the total kinetic energy of a rolling body becomes . The observation here that a rolling wheel has both kinds is the whole idea behind the classic problem of a sphere, a cylinder and a ring racing down an incline — a recurring JEE Main question, decided entirely by how the energy divides between the two forms.
Where the vibrational form leads. It becomes the kinetic theory of gases in Class 11, where the internal energy of a gas is the total kinetic energy of its molecules and temperature measures the average. The sentence in this page that heat is the vibrational kinetic energy of particles is the bridge NEET expects you to be able to state.
Where the energy-conversion chains lead. They reappear in Class 12 as efficiency questions and in Chemistry as thermochemistry, where chemical energy becomes heat with a measurable enthalpy change.
Question types to expect. At this level: and numericals, the free-fall verification, pendulum reasoning, and naming energy conversions. In competitive papers: energy conservation on tracks and loops, spring potential energy, rolling bodies, collisions, and kinetic theory.
The single trap that costs marks. Forgetting that the speed is squared. Doubling the speed quadruples the kinetic energy, and in JEE the same slip appears as taking the stopping distance to be proportional to the speed rather than to its square.
A second trap. Applying conservation of mechanical energy where friction is present. Mechanical energy is conserved only when no non-conservative force does work, so a block sliding down a rough incline needs the frictional work subtracted. Total energy is always conserved; mechanical energy is not — and stating which one you are using is what a competitive answer needs.
Board versus competitive emphasis. The ICSE paper marks the derivation, the substitution, the conservation statement and the named conversions; a competitive paper marks a speed or a height reached through an energy equation. **The transferable habit is writing at two chosen points and equating them** — one line that replaces a page of kinematics, and it works at every level from a dropped ball to a satellite.
**Where leads. Class 11 generalises it to any conservative force**, defining potential energy through , and then applies it to the spring, where . The fact you meet here — that gravitational potential energy depends on the height and not the path — is precisely the definition of a conservative force, and JEE questions turn on recognising when that is available.
**Where leads.** The relation that you verify here becomes essential in Collisions, where momentum and energy are used together, and again in Class 12 Dual Nature of Radiation for the de Broglie wavelength of a particle of given kinetic energy. NEET uses the momentum-energy relation directly.
Where the free-fall proof leads. It becomes the standard method for any problem involving heights and speeds without times: a block sliding down a curved track, a bead on a wire, a body on a loop. **JEE Advanced sets problems on vertical circular motion that are solved by writing at two points and equating them, which is the proof of this section applied twice.
Where the three forms of kinetic energy lead.** Class 11 quantifies the rotational part as , and the total kinetic energy of a rolling body becomes . The observation here that a rolling wheel has both kinds is the whole idea behind the classic problem of a sphere, a cylinder and a ring racing down an incline — a recurring JEE Main question, decided entirely by how the energy divides between the two forms.
Where the vibrational form leads. It becomes the kinetic theory of gases in Class 11, where the internal energy of a gas is the total kinetic energy of its molecules and temperature measures the average. The sentence in this page that heat is the vibrational kinetic energy of particles is the bridge NEET expects you to be able to state.
Where the energy-conversion chains lead. They reappear in Class 12 as efficiency questions and in Chemistry as thermochemistry, where chemical energy becomes heat with a measurable enthalpy change.
Question types to expect. At this level: and numericals, the free-fall verification, pendulum reasoning, and naming energy conversions. In competitive papers: energy conservation on tracks and loops, spring potential energy, rolling bodies, collisions, and kinetic theory.
The single trap that costs marks. Forgetting that the speed is squared. Doubling the speed quadruples the kinetic energy, and in JEE the same slip appears as taking the stopping distance to be proportional to the speed rather than to its square.
A second trap. Applying conservation of mechanical energy where friction is present. Mechanical energy is conserved only when no non-conservative force does work, so a block sliding down a rough incline needs the frictional work subtracted. Total energy is always conserved; mechanical energy is not — and stating which one you are using is what a competitive answer needs.
Board versus competitive emphasis. The ICSE paper marks the derivation, the substitution, the conservation statement and the named conversions; a competitive paper marks a speed or a height reached through an energy equation. **The transferable habit is writing at two chosen points and equating them** — one line that replaces a page of kinematics, and it works at every level from a dropped ball to a satellite.
Key takeaways
What must you be able to do from this part?
Two formulas, three kinds of kinetic energy and one principle with no exceptions.
- **, derived as the work done in lifting the body, measured from a stated reference level and independent of the path
- **, derived from with — and the acceleration cancels
- **A kg body at m has J**; a kg body at m s has J
- Doubling the speed quadruples the kinetic energy — J at m s
- ****, so a kg body with J has m s and kg m s
- Translational kinetic energy belongs to a body moving from place to place; rotational to one turning about an axis; vibrational to one moving to and fro
- A rolling wheel has both translational and rotational kinetic energy, and only the translational part is calculated at this level
- Heat is the vibrational kinetic energy of a body's particles
- Forms of energy: mechanical, chemical, heat, electrical, nuclear, sound and light
- Conversions to know: chemical to electrical in a cell, electrical to heat in an iron, electrical to mechanical in a fan, mechanical to electrical in a dynamo, light to electrical in a solar cell, sound to electrical in a microphone, heat to mechanical in a steam engine
- Energy is never destroyed — "lost" energy has become low-grade heat in the surroundings
- Conservation of energy: the total energy of an isolated system is constant; energy can be converted but neither created nor destroyed
- **For a free fall from height , at a height **: , , and with the cancelling
- **A kg ball dropped from m** has J at every point, split as J and J at the halfway height, and lands at m s
- Potential and kinetic energies are equal at half the original height
- A pendulum has maximum potential energy and zero kinetic energy at the extremes, and the reverse at the lowest point
- **A bob raised m reaches m s at the lowest point, whatever its mass
- A real pendulum stops because mechanical energy is converted into heat and sound, not because energy is lost
- The mass cancels in **, so all bodies dropped from the same height land at the same speed when air resistance is ignored
The best self-test costs nothing but a coin and a table. Drop it from the table top, work out the speed it should hit the floor at from the height alone, and then decide what measurement you would need to make to check whether air resistance mattered at all.
- **, derived as the work done in lifting the body, measured from a stated reference level and independent of the path
- **, derived from with — and the acceleration cancels
- **A kg body at m has J**; a kg body at m s has J
- Doubling the speed quadruples the kinetic energy — J at m s
- ****, so a kg body with J has m s and kg m s
- Translational kinetic energy belongs to a body moving from place to place; rotational to one turning about an axis; vibrational to one moving to and fro
- A rolling wheel has both translational and rotational kinetic energy, and only the translational part is calculated at this level
- Heat is the vibrational kinetic energy of a body's particles
- Forms of energy: mechanical, chemical, heat, electrical, nuclear, sound and light
- Conversions to know: chemical to electrical in a cell, electrical to heat in an iron, electrical to mechanical in a fan, mechanical to electrical in a dynamo, light to electrical in a solar cell, sound to electrical in a microphone, heat to mechanical in a steam engine
- Energy is never destroyed — "lost" energy has become low-grade heat in the surroundings
- Conservation of energy: the total energy of an isolated system is constant; energy can be converted but neither created nor destroyed
- **For a free fall from height , at a height **: , , and with the cancelling
- **A kg ball dropped from m** has J at every point, split as J and J at the halfway height, and lands at m s
- Potential and kinetic energies are equal at half the original height
- A pendulum has maximum potential energy and zero kinetic energy at the extremes, and the reverse at the lowest point
- **A bob raised m reaches m s at the lowest point, whatever its mass
- A real pendulum stops because mechanical energy is converted into heat and sound, not because energy is lost
- The mass cancels in **, so all bodies dropped from the same height land at the same speed when air resistance is ignored
The best self-test costs nothing but a coin and a table. Drop it from the table top, work out the speed it should hit the floor at from the height alone, and then decide what measurement you would need to make to check whether air resistance mattered at all.