A Fielder Pulls His Hands Back and the Catch Stops Hurting
Learn to calculate momentum and its change, derive F = ma from the rate of change of momentum, convert between the newton and the kilogram-force, and find the action-reaction pair in any situation.
Why does pulling your hands back make a catch painless?
A cricket ball of mass kg comes at a fielder at . Whether the fielder holds the hands rigid or draws them back, the ball has to be brought to rest — the change in motion is the same either way.
What differs is the time taken. And the force needed depends on that time.
Stopping the ball in s:
Stopping it in s by drawing the hands back:
One fifth of the force, for the same catch. Five times the stopping time, one fifth the force — the two are in exact inverse proportion.
The quantity that had to be destroyed in both cases is , and it is called momentum. The previous part of this chapter said that a force changes motion; this page says how much force changes it how fast. That relationship is Newton's second law, and it gives as a consequence rather than as a starting point.
This page covers the second part of the ICSE Class 9 Physics chapter on the laws of motion — momentum, the second law, the newton and the kilogram-force, and the third law.
What differs is the time taken. And the force needed depends on that time.
Stopping the ball in s:
Stopping it in s by drawing the hands back:
One fifth of the force, for the same catch. Five times the stopping time, one fifth the force — the two are in exact inverse proportion.
The quantity that had to be destroyed in both cases is , and it is called momentum. The previous part of this chapter said that a force changes motion; this page says how much force changes it how fast. That relationship is Newton's second law, and it gives as a consequence rather than as a starting point.
This page covers the second part of the ICSE Class 9 Physics chapter on the laws of motion — momentum, the second law, the newton and the kilogram-force, and the third law.
What is momentum and how do you find its change?
Momentum is the product of a body's mass and its velocity — a measure of how much motion it carries.
The SI unit is , which can also be written . Momentum is a vector, pointing along the velocity.
Worked example 1. A kg ball at :
Worked example 2. A kg car at :
Worked example 3 — same momentum, very different bodies. A truck of kg at also has
the same as the car above. A heavy slow body and a light fast one can carry identical momentum, and stopping either in the same time needs the same force.
Change in momentum is final minus initial:
Worked example 4 — slowing down. A kg body slows from to :
The minus sign says the change opposed the original motion.
Worked example 5 — a ball struck back. A cricket ball of kg arrives at and is struck straight back at . Taking the incoming direction as positive, the final velocity is :
so the magnitude of the change is .
**The change is , not .** Subtracting the speeds as and multiplying by gives , which would be right only if the ball carried on in the same direction. A reversal means the velocities have opposite signs, so their magnitudes ADD — and this is the single commonest error in momentum questions.
Worked example 6 — a body brought to rest. The same ball stopped dead by a fielder:
Momentum is not the same as inertia. Inertia depends only on mass and is the same whether a body moves or not; momentum depends on mass and velocity, and a stationary truck has enormous inertia and zero momentum. So "heavier bodies have more momentum" is false as stated — a stationary lorry has less momentum than a thrown cricket ball.
The SI unit is , which can also be written . Momentum is a vector, pointing along the velocity.
Worked example 1. A kg ball at :
Worked example 2. A kg car at :
Worked example 3 — same momentum, very different bodies. A truck of kg at also has
the same as the car above. A heavy slow body and a light fast one can carry identical momentum, and stopping either in the same time needs the same force.
Change in momentum is final minus initial:
Worked example 4 — slowing down. A kg body slows from to :
The minus sign says the change opposed the original motion.
Worked example 5 — a ball struck back. A cricket ball of kg arrives at and is struck straight back at . Taking the incoming direction as positive, the final velocity is :
so the magnitude of the change is .
**The change is , not .** Subtracting the speeds as and multiplying by gives , which would be right only if the ball carried on in the same direction. A reversal means the velocities have opposite signs, so their magnitudes ADD — and this is the single commonest error in momentum questions.
Worked example 6 — a body brought to rest. The same ball stopped dead by a fielder:
Momentum is not the same as inertia. Inertia depends only on mass and is the same whether a body moves or not; momentum depends on mass and velocity, and a stationary truck has enormous inertia and zero momentum. So "heavier bodies have more momentum" is false as stated — a stationary lorry has less momentum than a thrown cricket ball.
Formula
How do you get F = ma from the second law?
Newton's second law says the rate of change of momentum is proportional to the applied force and takes place in the direction of the force. The familiar follows in three lines.
The derivation. A body of mass has its velocity changed from to in time by a force . Its momentum changes from to , so
But is the acceleration , so this is simply . The second law says the force is proportional to this:
**The SI unit is chosen to make .** One newton is defined as the force that gives a mass of kg an acceleration of , which fixes at exactly one and leaves
**So is not the law — it is the law with a convenient unit. The momentum form is the more general one, and it is the one that handles a catch, a collision or a rocket, where the two forms part company at higher levels.
Worked example 1 — force from an acceleration.** A kg body speeds up from to in s:
Check by the momentum route: , so N. The two forms agree, as they must.
Worked example 2 — acceleration from a force. A force of N acts on a kg body:
Worked example 3 — the catch, both ways. The kg ball at has in magnitude. Stopped in s, N; stopped in s, N.
Worked example 4 — a retarding force. A kg car at is brought to rest in s:
The minus sign means the force of N acts against the motion.
Worked example 5 — finding the mass. A force of N produces an acceleration of :
Worked example 6 — a struck ball. A bat applies a force for s to the ball of worked example 5 in the previous section, whose momentum changed by :
** needs the NET force, not any single one.** A box pushed with N against N of friction accelerates according to the net N, so a kg box gets
**Using the N would give **, which is wrong — and the first law is what says why: only the unbalanced part of the force does anything.
The derivation. A body of mass has its velocity changed from to in time by a force . Its momentum changes from to , so
But is the acceleration , so this is simply . The second law says the force is proportional to this:
**The SI unit is chosen to make .** One newton is defined as the force that gives a mass of kg an acceleration of , which fixes at exactly one and leaves
**So is not the law — it is the law with a convenient unit. The momentum form is the more general one, and it is the one that handles a catch, a collision or a rocket, where the two forms part company at higher levels.
Worked example 1 — force from an acceleration.** A kg body speeds up from to in s:
Check by the momentum route: , so N. The two forms agree, as they must.
Worked example 2 — acceleration from a force. A force of N acts on a kg body:
Worked example 3 — the catch, both ways. The kg ball at has in magnitude. Stopped in s, N; stopped in s, N.
Worked example 4 — a retarding force. A kg car at is brought to rest in s:
The minus sign means the force of N acts against the motion.
Worked example 5 — finding the mass. A force of N produces an acceleration of :
Worked example 6 — a struck ball. A bat applies a force for s to the ball of worked example 5 in the previous section, whose momentum changed by :
** needs the NET force, not any single one.** A box pushed with N against N of friction accelerates according to the net N, so a kg box gets
**Using the N would give **, which is wrong — and the first law is what says why: only the unbalanced part of the force does anything.
How do you convert between the newton and the kilogram-force?
One kilogram-force is the weight of one kilogram, so it equals newtons:
The newton is the absolute or SI unit of force. The kilogram-force, also written kgf or kg wt, is a gravitational unit — defined by what gravity does to a kilogram.
Worked conversions.
- N
- N
-
-
-
Worked example — a person's weight in both units. A person of mass kg has weight
Notice the pattern: the weight in kgf is numerically the same as the mass in kg. That is exactly why the gravitational unit was invented — it lets a shopkeeper's scale read the same number for mass and for weight.
And it is also why the unit causes confusion. **A bag "weighing kg" has a mass of kg and a weight of kgf, which is N. Everyday speech uses the mass unit for a weight, and physics keeps them apart.
The corresponding CGS unit.** The gram-force is the weight of one gram, so
taking , and dyne, using N dyne from the measurements chapter.
The kilogram-force is not a fixed amount of force. Its size depends on , so it is slightly different at the equator and at a pole, and on the Moon a kilogram-force would be about a sixth of its value on Earth. The newton has no such problem, because it is defined from mass, length and time and not from gravity. That is the reason the SI system uses it, and the reason an answer should be converted to newtons before any calculation involving .
The newton is the absolute or SI unit of force. The kilogram-force, also written kgf or kg wt, is a gravitational unit — defined by what gravity does to a kilogram.
Worked conversions.
- N
- N
-
-
-
Worked example — a person's weight in both units. A person of mass kg has weight
Notice the pattern: the weight in kgf is numerically the same as the mass in kg. That is exactly why the gravitational unit was invented — it lets a shopkeeper's scale read the same number for mass and for weight.
And it is also why the unit causes confusion. **A bag "weighing kg" has a mass of kg and a weight of kgf, which is N. Everyday speech uses the mass unit for a weight, and physics keeps them apart.
The corresponding CGS unit.** The gram-force is the weight of one gram, so
taking , and dyne, using N dyne from the measurements chapter.
The kilogram-force is not a fixed amount of force. Its size depends on , so it is slightly different at the equator and at a pole, and on the Moon a kilogram-force would be about a sixth of its value on Earth. The newton has no such problem, because it is defined from mass, length and time and not from gravity. That is the reason the SI system uses it, and the reason an answer should be converted to newtons before any calculation involving .
Why do action and reaction not cancel each other out?
Because they act on two different bodies. Forces can only cancel if they act on the same body, and an action-reaction pair never does.
Newton's third law: to every action there is an equal and opposite reaction.
Worked example 1 — walking. Your foot pushes the ground backward; the ground pushes your foot forward. The backward push acts on the Earth and the forward push acts on you, so they appear in two separate force diagrams and nothing cancels. The force on you is what moves you.
Worked example 2 — rowing a boat. The oar pushes the water backward; the water pushes the oar, and so the boat, forward.
Worked example 3 — a rocket. Hot gases are pushed downward out of the nozzle; the gases push the rocket upward.
Worked example 4 — a book on a table. The book presses down on the table; the table presses up on the book with an equal force. These two are an action-reaction pair, and they do not cancel — one acts on the table and one on the book. The pair that cancels for the book is a different pair: the table's upward push and the Earth's downward pull, both acting on the book. Telling those two pairs apart is the heart of this topic.
Worked example 5 — recoil of a gun. A gun of mass kg fires a bullet of mass kg at . The forward and backward forces are equal and act for the same time, so the momentum gained by the bullet equals the momentum gained by the gun, in the opposite direction:
The gun recoils at . Equal momentum, unequal speed — the gun is times heavier, so it moves times more slowly.
Worked example 6 — jumping off a boat. A man of kg jumps from a boat of mass kg at :
The boat moves back at .
Equal forces do not mean equal accelerations. In the gun problem both bodies feel the same force for the same time, and their accelerations are in inverse ratio to their masses because . A mosquito striking a bus windscreen exerts exactly the same force on the bus as the bus does on it — the outcomes differ entirely because the masses do, not because the forces do.
Why the Earth does not visibly move when you walk. Your foot really does push the Earth backward with the same force that pushes you forward. The Earth's mass is so enormous that the resulting acceleration is far too small to detect. The force is equal; only the effect is unequal, and that single sentence answers most third-law questions.
Newton's third law: to every action there is an equal and opposite reaction.
Worked example 1 — walking. Your foot pushes the ground backward; the ground pushes your foot forward. The backward push acts on the Earth and the forward push acts on you, so they appear in two separate force diagrams and nothing cancels. The force on you is what moves you.
Worked example 2 — rowing a boat. The oar pushes the water backward; the water pushes the oar, and so the boat, forward.
Worked example 3 — a rocket. Hot gases are pushed downward out of the nozzle; the gases push the rocket upward.
Worked example 4 — a book on a table. The book presses down on the table; the table presses up on the book with an equal force. These two are an action-reaction pair, and they do not cancel — one acts on the table and one on the book. The pair that cancels for the book is a different pair: the table's upward push and the Earth's downward pull, both acting on the book. Telling those two pairs apart is the heart of this topic.
Worked example 5 — recoil of a gun. A gun of mass kg fires a bullet of mass kg at . The forward and backward forces are equal and act for the same time, so the momentum gained by the bullet equals the momentum gained by the gun, in the opposite direction:
The gun recoils at . Equal momentum, unequal speed — the gun is times heavier, so it moves times more slowly.
Worked example 6 — jumping off a boat. A man of kg jumps from a boat of mass kg at :
The boat moves back at .
Equal forces do not mean equal accelerations. In the gun problem both bodies feel the same force for the same time, and their accelerations are in inverse ratio to their masses because . A mosquito striking a bus windscreen exerts exactly the same force on the bus as the bus does on it — the outcomes differ entirely because the masses do, not because the forces do.
Why the Earth does not visibly move when you walk. Your foot really does push the Earth backward with the same force that pushes you forward. The Earth's mass is so enormous that the resulting acceleration is far too small to detect. The force is equal; only the effect is unequal, and that single sentence answers most third-law questions.
Exam tip
Exam tip: subtract momentum with signs, and name both bodies in a pair
**Change in momentum is with SIGNS.** A ball arriving at and struck back at changes by , not — the reversal makes the magnitudes add.
State the positive direction before writing any momentum, then keep it.
**Give the unit as ** (or ), never kg or .
**Derive when asked**: write the rate of change of momentum as , identify as , then say the SI unit is chosen to make the constant one. All three steps carry marks.
**Use the NET force in ** — N against N friction gives N, so a kg box gets .
A retarding force comes out negative, and you report its magnitude: a force of N opposing the motion.
Convert kgf before calculating: N, so N and N .
Weight in kgf is numerically the mass in kg — a kg person weighs N.
For a third-law pair, name BOTH bodies: the foot pushes the ground backward; the ground pushes the foot forward. A pair without two named bodies loses the mark.
Say why they do not cancel — they act on different bodies, and cancellation needs the same body.
And for recoil, equate the momenta: gives .
State the positive direction before writing any momentum, then keep it.
**Give the unit as ** (or ), never kg or .
**Derive when asked**: write the rate of change of momentum as , identify as , then say the SI unit is chosen to make the constant one. All three steps carry marks.
**Use the NET force in ** — N against N friction gives N, so a kg box gets .
A retarding force comes out negative, and you report its magnitude: a force of N opposing the motion.
Convert kgf before calculating: N, so N and N .
Weight in kgf is numerically the mass in kg — a kg person weighs N.
For a third-law pair, name BOTH bodies: the foot pushes the ground backward; the ground pushes the foot forward. A pair without two named bodies loses the mark.
Say why they do not cancel — they act on different bodies, and cancellation needs the same body.
And for recoil, equate the momenta: gives .
Did you know
Why a mosquito and a bus hit each other equally hard
A mosquito flies into the windscreen of a moving bus. The mosquito is destroyed and the bus is unmarked.
Ask which one exerted the greater force and the honest answer is neither — the two forces were exactly equal and opposite. The third law leaves no room for negotiation about it.
What differs is not the force but what the force does, and that is decided by the mass. Since , the same force produces an enormous acceleration on a body of a few milligrams and an utterly negligible one on a body of several tonnes.
Put rough numbers to it. Suppose the impact force is on each. A mosquito of mass kg and a bus of kg experience accelerations in the ratio
Four thousand million times the acceleration, from the identical force.
The momentum change tells the same story from the other side. The mosquito and the bus receive equal and opposite changes in momentum, because the forces are equal and act for the same duration. The bus's velocity changes by an amount far too small to measure, and the mosquito's changes completely — same momentum transfer, wildly different fractions of what each body already had.
The gun and bullet in the last section are the same arrangement with less extreme masses, and there the recoil is noticeable: backward against forward, in the ratio of to because the masses are in the ratio of to .
So the third law is not the claim that collisions are fair. It is the claim that the forces are equal, and the second law then decides, mass by mass, who comes off worse.
Ask which one exerted the greater force and the honest answer is neither — the two forces were exactly equal and opposite. The third law leaves no room for negotiation about it.
What differs is not the force but what the force does, and that is decided by the mass. Since , the same force produces an enormous acceleration on a body of a few milligrams and an utterly negligible one on a body of several tonnes.
Put rough numbers to it. Suppose the impact force is on each. A mosquito of mass kg and a bus of kg experience accelerations in the ratio
Four thousand million times the acceleration, from the identical force.
The momentum change tells the same story from the other side. The mosquito and the bus receive equal and opposite changes in momentum, because the forces are equal and act for the same duration. The bus's velocity changes by an amount far too small to measure, and the mosquito's changes completely — same momentum transfer, wildly different fractions of what each body already had.
The gun and bullet in the last section are the same arrangement with less extreme masses, and there the recoil is noticeable: backward against forward, in the ratio of to because the masses are in the ratio of to .
So the third law is not the claim that collisions are fair. It is the claim that the forces are equal, and the second law then decides, mass by mass, who comes off worse.
Exam relevance
How are momentum and the second law tested in JEE Main and NEET?
Because momentum is conserved in every collision, and the momentum form of the second law is the one that survives into higher physics.
This is the foundation for Class 11 Physics Laws of Motion and Systems of Particles and Rotational Motion, examined in JEE Main and NEET. The second law is written there as
which is this page's rate of change of momentum with the rate taken instantaneously. ** is then shown to be the special case where the mass is constant — and questions on a rocket losing mass, or a conveyor belt gaining it, exist precisely because that assumption fails there. The distinction drawn on this page, that the momentum form is the more general one, is what makes those problems tractable.
Conservation of momentum** is the third law's main consequence and a recurring exam topic. Because action and reaction are equal and opposite and act for the same time, the momentum changes cancel and the total momentum of an isolated system stays constant:
The gun-recoil and boat-jumping calculations here are that equation with one side starting from rest. Class 11 extends it to collisions in one and two dimensions, and elastic against inelastic becomes the new distinction.
Impulse is the formal version of the catch. Class 11 defines impulse as and shows it equals the change in momentum, so
makes the fielder's trick a one-line argument: fix , stretch , and falls. Impulse-momentum numericals are standard in both exams, often about airbags, crumple zones or a long jumper landing on sand — all the same physics as drawing the hands back.
The equal-forces-unequal-accelerations point is tested as an assertion-reason item in both papers, usually with a heavy and a light body colliding.
Where the kilogram-force disappears. Competitive papers work exclusively in newtons, so the gravitational unit is a board-level convention. Converting to newtons before starting is the habit that transfers.
What the questions look like. For board work, expect calculate momentum and its change, **derive from the second law, numericals on force, mass and change in velocity, convert between newton and kgf, identify an action-reaction pair, and explain why they do not cancel. For JEE Main and NEET, expect impulse and momentum-conservation numericals, collision problems, and free-body diagrams with the net force.
How board and competitive emphasis differ. A board paper rewards the written derivation and a pair named with both bodies. A competitive paper assumes all of it and tests conservation of momentum in a collision — where the marks turn on the signs, not on the concepts.
The single trap that costs the most marks.** Subtracting speeds instead of velocities when a body reverses. A ball arriving at and returning at has a momentum change of , and treating it as gives a plausible number that is five times too small. **The defence is to write the final velocity as explicitly** before substituting, so the sign cannot be lost in the arithmetic.
This is the foundation for Class 11 Physics Laws of Motion and Systems of Particles and Rotational Motion, examined in JEE Main and NEET. The second law is written there as
which is this page's rate of change of momentum with the rate taken instantaneously. ** is then shown to be the special case where the mass is constant — and questions on a rocket losing mass, or a conveyor belt gaining it, exist precisely because that assumption fails there. The distinction drawn on this page, that the momentum form is the more general one, is what makes those problems tractable.
Conservation of momentum** is the third law's main consequence and a recurring exam topic. Because action and reaction are equal and opposite and act for the same time, the momentum changes cancel and the total momentum of an isolated system stays constant:
The gun-recoil and boat-jumping calculations here are that equation with one side starting from rest. Class 11 extends it to collisions in one and two dimensions, and elastic against inelastic becomes the new distinction.
Impulse is the formal version of the catch. Class 11 defines impulse as and shows it equals the change in momentum, so
makes the fielder's trick a one-line argument: fix , stretch , and falls. Impulse-momentum numericals are standard in both exams, often about airbags, crumple zones or a long jumper landing on sand — all the same physics as drawing the hands back.
The equal-forces-unequal-accelerations point is tested as an assertion-reason item in both papers, usually with a heavy and a light body colliding.
Where the kilogram-force disappears. Competitive papers work exclusively in newtons, so the gravitational unit is a board-level convention. Converting to newtons before starting is the habit that transfers.
What the questions look like. For board work, expect calculate momentum and its change, **derive from the second law, numericals on force, mass and change in velocity, convert between newton and kgf, identify an action-reaction pair, and explain why they do not cancel. For JEE Main and NEET, expect impulse and momentum-conservation numericals, collision problems, and free-body diagrams with the net force.
How board and competitive emphasis differ. A board paper rewards the written derivation and a pair named with both bodies. A competitive paper assumes all of it and tests conservation of momentum in a collision — where the marks turn on the signs, not on the concepts.
The single trap that costs the most marks.** Subtracting speeds instead of velocities when a body reverses. A ball arriving at and returning at has a momentum change of , and treating it as gives a plausible number that is five times too small. **The defence is to write the final velocity as explicitly** before substituting, so the sign cannot be lost in the arithmetic.
Key takeaways
Momentum, the second law and the third law: quick revision
- Momentum , in (or ), a vector along the velocity.
- A kg ball at has ; a kg car at has ; so does a kg truck at .
- Change in momentum , with signs.
- A kg body slowing from to : .
- A ball reversed — kg arriving at and struck back at — changes by ****, not : the magnitudes add.
- Momentum is not inertia — a stationary truck has huge inertia and zero momentum.
- Second law: the rate of change of momentum is proportional to the force and acts in its direction.
- Derivation: , and the SI unit is defined to make the constant , giving **** and .
- **One newton gives kg an acceleration of .**
- A kg body from to in s needs N, by either route.
- N on kg gives ; N producing means kg; kg stopped from in s needs N against the motion.
- The catch: gives N in s and only N in s — five times the time, one fifth the force.
- A bat acting for s on a change exerts N.
- Use the NET force: N against N friction on a kg box gives , not .
- ** N**: N, N, N , N .
- A kg person weighs N — the kgf number equals the kg number, which is why the unit exists and why it confuses.
- **The kgf depends on and the newton does not**, so convert to newtons before using . Also dyne.
- Third law: to every action there is an equal and opposite reaction, acting on a different body.
- Walking, rowing, a rocket, a book on a table — name both bodies in each pair.
- They do not cancel because cancellation needs the SAME body. For the book, the cancelling pair is its weight and the table's push, both on the book.
- Recoil: a kg gun firing a kg bullet at recoils at .
- Boat: a kg man jumping at pushes a kg boat back at .
- Equal forces, unequal accelerations — a mosquito and a bus push each other equally, and decides the outcome.
Work out the momentum of a cricket ball bowled at you and of a lorry crawling in traffic, and see which one you would rather stop in a tenth of a second.
- A kg ball at has ; a kg car at has ; so does a kg truck at .
- Change in momentum , with signs.
- A kg body slowing from to : .
- A ball reversed — kg arriving at and struck back at — changes by ****, not : the magnitudes add.
- Momentum is not inertia — a stationary truck has huge inertia and zero momentum.
- Second law: the rate of change of momentum is proportional to the force and acts in its direction.
- Derivation: , and the SI unit is defined to make the constant , giving **** and .
- **One newton gives kg an acceleration of .**
- A kg body from to in s needs N, by either route.
- N on kg gives ; N producing means kg; kg stopped from in s needs N against the motion.
- The catch: gives N in s and only N in s — five times the time, one fifth the force.
- A bat acting for s on a change exerts N.
- Use the NET force: N against N friction on a kg box gives , not .
- ** N**: N, N, N , N .
- A kg person weighs N — the kgf number equals the kg number, which is why the unit exists and why it confuses.
- **The kgf depends on and the newton does not**, so convert to newtons before using . Also dyne.
- Third law: to every action there is an equal and opposite reaction, acting on a different body.
- Walking, rowing, a rocket, a book on a table — name both bodies in each pair.
- They do not cancel because cancellation needs the SAME body. For the book, the cancelling pair is its weight and the table's push, both on the book.
- Recoil: a kg gun firing a kg bullet at recoils at .
- Boat: a kg man jumping at pushes a kg boat back at .
- Equal forces, unequal accelerations — a mosquito and a bus push each other equally, and decides the outcome.
Work out the momentum of a cricket ball bowled at you and of a lorry crawling in traffic, and see which one you would rather stop in a tenth of a second.