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

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.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2019 · Q55Moderate

Example 1 · Laws of Motion and Forces · Impulse and Momentum

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

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

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.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2025 · Q68Easy

Example 2 · Laws of Motion and Forces · Impulse and Momentum

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

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 4 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

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