NDA Physics · Laws of Motion and Forces

Impulse and Momentum

Momentum p = mv is the 'quantity of motion'; impulse is the product of force and the time it acts, and equals the change in momentum it produces.

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Why this matters

A compact, high-yield subtopic — roughly 5 PYQs across 2019–2026. The two ideas are p = mv and impulse = Ft = change in momentum, plus the everyday cushioning principle (pulling hands back, jumping onto sand) that all derive from spreading a momentum change over a longer time to cut the force. One formula and one principle clear the whole subtopic.

Concept 1 of 2

Linear momentum, p = mv

Intuition

Momentum captures how hard a body is to stop — it grows with both mass and velocity. A slow truck and a fast bullet can carry the same momentum. Because velocity is a vector, momentum is a vector too: it has direction, and reversing direction changes the momentum even if the speed is unchanged.

Definition

Linear momentum is the product of a body's mass and velocity, $\vec{p} = m\vec{v}$ . It is a vector pointing along the velocity, with SI unit kg m/s. Because it is a vector, a change in DIRECTION changes the momentum even at constant speed — this is why a bouncing ball's momentum changes while its speed (and kinetic energy) need not.

Linear momentum

\vec{p} = m\vec{v}

plinear momentum (kg m/s), a vector
mmass (kg)
vvelocity (m/s), a vector

Worked example

A 2 kg ball moving at 5 m/s strikes a wall and bounces straight back at 5 m/s. Find the change in its momentum.

Take the initial direction as positive: initial momentum $p_i = 2 \times 5 = +10\,\text{kg m/s}$ .
After bouncing it moves the opposite way at the same speed: $p_f = 2 \times (-5) = -10\,\text{kg m/s}$ .
Change $\Delta p = p_f - p_i = -10 - 10 = -20\,\text{kg m/s}$ .

Answer:20 kg m/s, directed away from the wall (the magnitude is 2mv, not zero).

Practice this conceptself-check · 4 quick reps

Try it yourself

When a ball bounces elastically off the ground (no energy lost), which quantity changes suddenly: speed, momentum, or kinetic energy?

Practice — Level 1 (4 reps)

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

1.
What is the formula for linear momentum?
2.
Momentum of a 3 kg body at 4 m/s?
3.
Is momentum a scalar or a vector?
4.
A ball bounces back at the same speed. Does its momentum change?

From the bank · past-year question

Example 1Laws of Motion and ForcesMODERATE

When a ball bounces off the ground, which of the following changes suddenly? (Assume no loss of energy to the floor)

[Q55 · Sep · 2019]

On an elastic bounce, momentum changes but speed and KE don't

NDA 2019 asked what changes suddenly when a ball bounces with no energy loss. Speed and kinetic energy (scalars depending on speed only) are unchanged; MOMENTUM changes because its direction reverses. The change in momentum is 2mv, not zero.

Concept 2 of 2

Impulse = change in momentum; the cushioning principle

Intuition

Impulse is force multiplied by the time it acts, and it equals the change in momentum it produces. The key insight: to change a body's momentum by a fixed amount, a small force over a long time does the same job as a large force over a short time. So if you can stretch out the contact TIME, you cut the peak FORCE — this is why a fielder pulls his hands back and why landing on sand hurts less.

Definition

Impulse $J = F\,\Delta t$ equals the change in momentum: $F\,\Delta t = \Delta p = m(v - u)$ (the impulse-momentum theorem). Graphically, impulse is the area under a force-time graph. Since $F = \Delta p / \Delta t$ , increasing the contact time $\Delta t$ for the same $\Delta p$ reduces the force — the basis of all cushioning.

Impulse-momentum theorem

J = F\,\Delta t = \Delta p = m(v - u)

Jimpulse (N·s, equivalently kg m/s)
F(average) force
Δttime over which the force acts
Δp = m(v − u)change in momentum

Worked example

A 0.15 kg cricket ball arrives at 20 m/s and is brought to rest by a fielder. If he stops it in 0.5 s instead of 0.1 s, compare the average force in the two cases.

Change in momentum is the same either way: $\Delta p = m(v - u) = 0.15 \times (0 - 20) = -3\,\text{kg m/s}$ (magnitude 3 N·s).
Quick stop: $F = \Delta p / \Delta t = 3 / 0.1 = 30\,\text{N}$ .
Slow stop (hands pulled back): $F = 3 / 0.5 = 6\,\text{N}$ .
Five times the time means one-fifth the force.

Answer:30 N for the quick stop vs 6 N for the cushioned stop — stretching the time cuts the force fivefold.

Practice this conceptself-check · 4 quick reps

Try it yourself

The force on a 10 kg object follows a force-time graph that is a trapezoid: it rises linearly from 0 to 20 N over the first 5 s, stays at 20 N from 5 s to 20 s, then falls linearly back to 0 from 20 s to 25 s. Find the final speed (it started from rest).

Practice — Level 1 (4 reps)

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

1.
What does impulse equal?
2.
What is the SI unit of impulse?
3.
On a force-time graph, impulse is represented by what?
4.
Why does a fielder pull his hands back when catching a ball?

From the bank · past-year question

Example 2Laws of Motion and ForcesEASY

Why a fielder in the ground gradually pulls his hands backward while catching a fast moving cricket ball ?

[Q68 · Sep · 2025]

Cushioning increases TIME to reduce FORCE

Jumping onto sand, pulling hands back, crumple zones in cars, bending knees on landing — all work by increasing the impact TIME. The momentum change Δp is fixed; spreading it over a longer Δt lowers the peak force F = Δp/Δt. The reason is reduced force (reduced acceleration), not reduced momentum change.

Net force on the floor in a bounce includes weight

When a ball bounces, the floor's reaction must both reverse the ball's momentum AND support its weight. NDA 2024 (0.1 kg dropped from 0.45 m, rebounds to 0.20 m, contact 0.1 s): impact force = Δp/t = 0.1(3+2)/0.1 = 5 N, then add mg = 1 N to get 6 N net.

Drill 3 more on impulse = change in momentum; the cushioning principle

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 (2)

Linear momentum, p = mv
Linear momentum
$\vec{p} = m\vec{v}$
Impulse = change in momentum; the cushioning principle
Impulse-momentum theorem
$J = F\,\Delta t = \Delta p = m(v - u)$

Watch out for (3)

On an elastic bounce, momentum changes but speed and KE don't
→ Linear momentum, p = mv
Cushioning increases TIME to reduce FORCE
→ Impulse = change in momentum; the cushioning principle
Net force on the floor in a bounce includes weight
→ Impulse = change in momentum; the cushioning principle

Mastery check — 3 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

A ball of 0.1 kg mass is dropped from height 0.45 m and rises to height 0.20 m. In contact with floor for 0.1 s. The net force applied on the floor while bouncing is (g = 10 m

s^{-2}

) :

[Q86 · Apr · 2024]

Example 2Laws of Motion and ForcesMODERATE

The force-time (F-t) graph of a 10 kg object shows a trapezoid: F rises linearly to 20 N at t=5 s, constant 20 N from t=5 to t=20 s, drops to 0 at t=25 s. What is the final speed of the object?

[Q75 · Apr · 2026]

Example 3Laws of Motion and ForcesEASY

A person jumps from a height on soft sand. Which one is the correct reason for less likely injury?

[Q63 · Apr · 2026]

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