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SPI Physics · Laws of Motion and Forces

Momentum and Force

Momentum is mass times velocity; a force is what changes it, and the force equals how fast the momentum changes, while the total momentum of bodies that only push on each other stays the same.

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Why this matters

Three questions in the bank, one EASY and two MODERATE, one of them from the 2022 paper: the force from a change of momentum in a short time, the force needed to hold a machine gun steady, and two cars that stick together after a head-on crash. All three use one idea, momentum, so learn it with its direction.

Concept 1 of 3: Newton's three laws of motion

A ball on the floor stays put until you kick it, and a kick on a heavy ball moves it less than the same kick on a light one. When you kick, the ball pushes back on your foot. Those three everyday facts are Newton's three laws.

Definition

  • First law (inertia): a body stays at rest, or keeps moving in a straight line at the same speed, unless a force acts on it.
  • Second law: the force on a body equals its mass times its acceleration, F=maF = ma. The unit of force is the newton: 1 N gives 1 kg an acceleration of 1 m/s².
  • Third law: to every action there is an equal and opposite reaction. The two forces act on different bodies, so they do not cancel.
  • A gun recoils when it fires: it pushes the bullet forward and the bullet pushes it back.

Newton's second law

F=maF = ma

Worked example

A force acts on a 2 kg trolley and gives it an acceleration of 3 m/s². Find the force.
Practice this concept3 quick reps

Cancelling action against reaction

Forces cancel only when they act on the same body. Action and reaction act on two different bodies (the gun and the bullet), so each body still moves.

Concept 2 of 3: Momentum and force as its rate of change

A slow truck and a fast cricket ball are both hard to stop: one has a large mass, the other a large speed. Momentum puts the two together. Stopping it quickly needs a big force; stopping it slowly needs a small one.

Definition

  • Momentum p=mvp = mv. Its unit is kg m/s, and it has a direction, the direction of the velocity.
  • Force == change in momentum ÷\div time taken. The same change of momentum in a shorter time needs a larger force.
  • That is why a fielder draws his hands back while catching: the catch takes longer, so the force on his hands is smaller.
  • For a gun firing a stream of bullets, the force needed to hold it steady == (bullets fired per second) ×\times (momentum of one bullet). The gun's own mass does not come into it.

Momentum and force

p=mvF=ΔpΔtF=n×mv for n bullets each secondp = mv \qquad F = \frac{\Delta p}{\Delta t} \qquad F = n \times m v \text{ for } n \text{ bullets each second}

Worked example

A ball of mass 0.2 kg moving at 10 m/s is stopped by a fielder in 0.05 s. Find the force on his hands.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q82Moderate

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

A machine gun has mass 4kg. It fires 50g bullets at the rate of 30 bullets per minute at muzzle velocity of 400m/ sec. What force is required to keep gun in position?

Using bullets per minute

Force is momentum per SECOND. If the rate is given per minute, divide by 60 first: 90 bullets a minute is 1.5 a second.

Bringing the gun's mass into the force

The force to hold a firing gun steady is the momentum given to the bullets each second. The gun's mass is extra information here.

Leaving grams in the momentum

Mass must be in kilograms: a 30 g bullet is 0.03 kg. Using 30 makes the momentum a thousand times too large.
Drill 1 more on momentum and force as its rate of change

Concept 3 of 3: Conservation of momentum in collisions and recoil

When two bodies collide, each pushes the other equally hard for the same time, in opposite directions. So whatever momentum one gains, the other loses. Add up the momentum before and after, with directions, and the two totals match.

Definition

  • If no outside force acts, the total momentum of a group of bodies stays the same before and after they collide, explode or push apart.
  • Momentum has direction. Pick one direction as positive; momentum the other way is negative.
  • Two bodies moving towards each other: subtract their momenta. The joined bodies move in the direction of the larger one.
  • A gun and bullet at rest have zero total momentum, so after firing mgunvgun=mbulletvbulletm_{\text{gun}} v_{\text{gun}} = m_{\text{bullet}} v_{\text{bullet}}, in opposite directions: that is recoil.

Conservation of momentum

m1u1+m2u2=m1v1+m2v2m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

Worked example

A 5 kg gun fires a 25 g bullet at 300 m/s. Find the speed with which the gun recoils.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Laws of Motion and Forces · Momentum and Force

Two cars A and B approaching each other with a momentum of 50 kg m/s and 25 kg m/s respectively, stick to each other after the collision. The momentum after the collision is:

Adding momenta that point opposite ways

Bodies moving towards each other have momenta in opposite directions. Subtract them; adding gives a total that is too large.

Forgetting the direction of the answer

After two bodies stick, they move in the direction of the one that had the larger momentum.

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)

Watch out for (6)

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paperDrill all 3 questions