NDA Physics · Laws of Motion and Forces

Conservation of Momentum and Collisions

In the absence of external forces, total linear momentum is conserved; this governs recoil, collisions, and any system where mass is being added or ejected.

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Why this matters

Roughly 8 PYQs across 2019–2024 — the second-biggest computation pocket in the chapter. Every problem reduces to one rule: total momentum before = total momentum after. Recoil of a gun, a boy jumping onto a cart, sand on a conveyor belt, equal-mass elastic collisions — all are the same conservation statement with the algebra rearranged.

Concept 1 of 3

Conservation of linear momentum

Intuition

If no outside force pushes on a system, its total momentum cannot change — internal forces (like a chemical explosion or two balls colliding) only shuffle momentum between the parts. So before-and-after, the total stays the same. This single rule solves recoil, explosions, and 'stick-together' collisions.

Definition

Law of conservation of linear momentum: for a system with no net external force, the total momentum is constant: $\sum \vec{p}_{\text{before}} = \sum \vec{p}_{\text{after}}$ . Internal forces (explosions, collisions, chemical reactions) cannot change the total momentum or the velocity of the centre of mass. It follows directly from Newton's third law: internal action-reaction pairs cancel.

Conservation of momentum (two bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

m₁, m₂masses of the two bodies
u₁, u₂velocities before
v₁, v₂velocities after

Worked example

A 10 g bullet is fired at 300 m/s from a 1 kg pistol initially at rest. Find the recoil velocity of the pistol.

Total momentum before firing is zero (everything at rest).
Conservation: $0 = m_b v_b + m_p v_p$ , with $m_b = 0.01\,\text{kg}$ , $v_b = 300\,\text{m/s}$ , $m_p = 1\,\text{kg}$ .
$v_p = -\dfrac{0.01 \times 300}{1} = -3\,\text{m/s}$ .
The minus sign means the pistol recoils opposite to the bullet.

Answer:-3 m/s (3 m/s backward).

Practice this conceptself-check · 4 quick reps

Try it yourself

A 52 kg boy runs horizontally at 2 m/s and jumps onto a stationary 3 kg cart on frictionless wheels. They then move together. Find their common speed.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
When is total linear momentum conserved?
2.
Can internal forces change a system's total momentum?
3.
A 0.02 kg bullet leaves a 2 kg gun at 200 m/s. Recoil speed?
4.
Can a chemical reaction inside a body change its centre-of-mass velocity?

From the bank · past-year question

Example 1Laws of Motion and ForcesMODERATE

A bullet of mass 10 g is horizontally fired with velocity 300 m s

^{-1}

from a pistol of mass 1 kg. What is the recoil velocity of the pistol?

[Q149 · Sep · 2022]

Internal forces can't move the centre of mass

NDA 2019: an object moving at velocity v has a chemical reaction inside it. The reaction is internal, so it cannot change the velocity of the centre of mass (statement 1 correct). It CAN, however, redistribute kinetic energy among the particles, so the energy-conservation statement is false. Internal forces redistribute, never reset the total.

Drill 3 more on conservation of linear momentum

Concept 2 of 3

Force when mass is added or ejected (variable mass)

Intuition

When mass piles onto a moving object — sand falling onto a belt, rain filling a wagon — momentum must still be accounted for. Falling material that has no horizontal motion adds horizontal mass but no horizontal momentum, so the speed drops to keep momentum constant. To keep the belt moving steadily you must supply a force equal to the rate at which momentum is being added.

Definition

When mass is added or ejected, force is the rate of change of momentum: $F = \dfrac{dp}{dt} = v\,\dfrac{dm}{dt}$ when the added mass arrives with no momentum along the motion. If material falls vertically onto a horizontally moving body, horizontal momentum is conserved: $M_1 v_1 = M_2 v_2$ , so the body slows as it gains mass.

Force to maintain speed while loading mass at rate dm/dt

F = v\,\frac{dm}{dt}

Fforce needed to keep speed constant
v(constant) speed of the body
dm/dtrate at which mass is added

Worked example

Sand falls vertically onto a conveyor belt at 0.1 kg/s. The belt must keep moving at a steady 2 m/s. What horizontal force is needed on the belt?

The sand lands with zero horizontal speed and must be accelerated up to the belt speed.
Force = rate of change of momentum = $v\,\dfrac{dm}{dt}$ .
$F = 2 \times 0.1 = 0.2\,\text{N}$ .

Answer:0.2 N.

Practice this conceptself-check · 4 quick reps

Try it yourself

An open railway wagon of mass M₁ moves at speed v₁. Rain falls vertically into it until its mass is M₂ and its speed is v₂. The water is at rest horizontally inside. Relate v₁ and v₂.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Force to keep a belt at 3 m/s while sand lands at 0.2 kg/s?
2.
Rain falls vertically into a moving open wagon. What is conserved horizontally?
3.
As a wagon fills with vertically-falling rain, does its speed rise or fall?
4.
Why does vertically-falling rain add no horizontal momentum?

From the bank · past-year question

Example 2Laws of Motion and ForcesMODERATE

Sand falls vertically on a conveyor belt at a rate of 0·1 kg/s. In order to keep the belt moving at a uniform speed of 2 m/s, the force required to be applied on the belt is :

[Q122 · Apr · 2023]

Vertically-falling mass adds no horizontal momentum

NDA 2023 (rain into a moving wagon): the rain has zero horizontal speed, so horizontal momentum is conserved as M₁v₁ = M₂v₂ — the wagon SLOWS. Don't assume the speed stays the same; the added mass must be dragged up to speed, and with no external horizontal force the wagon pays for it by slowing.

Drill 1 more on force when mass is added or ejected (variable mass)

Concept 3 of 3

Collisions — elastic and the equal-mass result

Intuition

In a collision, momentum is always conserved. In an ELASTIC collision, kinetic energy is conserved too. The most-tested special case: when two equal masses collide head-on elastically and one was at rest, they simply swap velocities — the moving one stops dead and the struck one moves off with the full original speed.

Definition

In all collisions momentum is conserved. In an elastic collision kinetic energy is also conserved; in an inelastic collision some KE is lost (to heat/deformation), and in a perfectly inelastic collision the bodies stick and move together. Special case — equal masses, one at rest, elastic, head-on: the velocities are exchanged. The moving body stops and the target moves off with the incoming speed.

Equal-mass elastic head-on collision (m₂ initially at rest)

v_1' = 0, \qquad v_2' = u_1

u₁speed of the incoming body (mass m)
v₁'speed of body 1 after — it stops
v₂'speed of body 2 after — it takes the full speed

Worked example

A 0.2 kg ball moving at 6 m/s collides head-on with a stationary 0.3 kg ball. After the collision the 0.2 kg ball stops dead. Find the speed of the 0.3 kg ball.

Conserve momentum: $0.2 \times 6 = 0.2 \times 0 + 0.3 \times v$ .
$1.2 = 0.3 v$ .
$v = 1.2 / 0.3 = 4\,\text{m/s}$ .

Answer:4 m/s.

Practice this conceptself-check · 4 quick reps

Try it yourself

A bob X of mass m swings down and collides elastically, head-on, with an identical bob Y at rest on a frictionless surface. What happens to X immediately after the collision?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
What is conserved in every collision?
2.
What extra quantity is conserved in an elastic collision?
3.
Equal masses, elastic head-on, one at rest: what does the moving one do?
4.
In a perfectly inelastic collision, what happens to the bodies?

From the bank · past-year question

Example 3Laws of Motion and ForcesMODERATE

A 100 g sphere is moving at a speed of 20 m/s and collides with another sphere of mass 50 g. If the second sphere was at rest prior to the collision and the first sphere comes at rest immediately after the collision, considering the collision to be elastic, the speed of the second sphere would be

[Q133 · Sep · 2023]

Equal-mass elastic collision: velocities are EXCHANGED

NDA 2024 (bob X hits identical bob Y at rest): X does not bounce back or rise on the other side — it STOPS at the collision point and Y carries off all the speed. This swap only happens for equal masses in an elastic head-on hit; unequal masses share the velocity differently.

Use momentum (not KE) to find the unknown speed

When a problem tells you the post-collision state (e.g. the first sphere stops), use conservation of MOMENTUM to find the other speed. Plugging into kinetic-energy conservation is unnecessary and a common time-sink; momentum alone gives the answer directly.

Drill 1 more on collisions — elastic and the equal-mass result

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Conservation of linear momentum
Conservation of momentum (two bodies)
$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$
Force when mass is added or ejected (variable mass)
Force to maintain speed while loading mass at rate dm/dt
$F = v\,\frac{dm}{dt}$
Collisions — elastic and the equal-mass result
Equal-mass elastic head-on collision (m₂ initially at rest)
$v_1' = 0, \qquad v_2' = u_1$

Watch out for (4)

Internal forces can't move the centre of mass
→ Conservation of linear momentum
Vertically-falling mass adds no horizontal momentum
→ Force when mass is added or ejected (variable mass)
Equal-mass elastic collision: velocities are EXCHANGED
→ Collisions — elastic and the equal-mass result
Use momentum (not KE) to find the unknown speed
→ Collisions — elastic and the equal-mass result

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Laws of Motion and ForcesHARD

Let there be an object having some chemicals in it. It starts moving with a uniform velocity

v

and a chemical reaction starts happening. In this case, which of the following statement/s is/are correct? 1. Chemical reactions happening in the system cannot change the velocity

v

of the center of mass of the object. 2. Chemical reactions happening in the system cannot change kinetic energy of the particles inside with respect to the center of mass of object. Select the correct answer using the code given below:

[Q52 · Sep · 2019]

Example 2Laws of Motion and ForcesMODERATE

A railway wagon (open at the top) of mass

M_1

is moving with speed

v_1

along a straight track. Rain partially fills it so mass becomes

M_2

and speed becomes

v_2

. Taking rain to fall vertically and water stationary inside, the relation between

v_1

and

v_2

is :

[Q87 · Apr · 2023]

Example 3Laws of Motion and ForcesMODERATE

A metallic bob X of mass m is released from position A. It collides elastically with another identical bob Y placed at rest at position B on a horizontal frictionless table. The angle AOB is 30°. How high does the bob X rise immediately after the collision ?

[Q89 · Apr · 2024]

Example 4Laws of Motion and ForcesEASY

Fundamental laws of physics require

[Q116 · Sep · 2021]

Example 5Laws of Motion and ForcesMODERATE

A boy of mass 52 kg jumps with a horizontal velocity of 2 m/s onto a stationary cart of mass 3 kg. The cart is fixed with frictionless wheels. Which one of the following would be the speed of the cart?

[Q81 · Apr · 2022]

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