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4.5.5Forces and Motion

Collisions

5 min read 9 parts Lesson 37 of 66 1 quiz

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Lesson summary

4.5.5 Collisions — AQA A Level Physics 7408 notes: Apply conservation of momentum to elastic, inelastic and explosive interactions in one dimension and judge whether a collision is elastic using kinetic energy.The principle of conservation of momentum states that, for a system of…

  • elastic collision
  • perfectly inelastic
  • Worked example 1
  • Worked example 2
  • Worked example 3
  • Isolated system

Section 1 of 9Principle of conservation of momentum

The principle of conservation of momentum states that, for a system of interacting bodies, the total momentum before an interaction equals the total momentum after it, provided no external resultant force acts on the system.

Principle of conservation of momentum
During a collision the two bodies exert forces on each other that are equal and opposite (Newton's third law) and act for the same time.
The impulses are therefore equal and opposite, so the momentum gained by one body equals the momentum lost by the other.

In one dimension

  • m1u1 + m2u2 = m1v1 + m2v2.
  • Choose a positive direction first and give any velocity in the opposite direction a negative sign.

Section 2 of 9Elastic collisions

In an elastic collision kinetic energy is conserved as well as momentum. Total kinetic energy before = total kinetic energy after, so ½m1u12 + ½m2u22 = ½m1v12 + ½m2v22.

For an elastic collision in a straight line the relative speed of approach equals the relative speed of separation: u1 − u2 = v2 − v1.

Elastic collisions
Collisions between gas molecules and between hard steel balls are close to elastic.
Because momentum is a vector but energy a scalar, the kinetic energy test must use v2, not v.

Section 3 of 9Inelastic collisions

In an inelastic collision momentum is conserved but some kinetic energy is transferred to other forms, such as thermal energy, sound and energy of deformation.

The total kinetic energy after is less than before.
Perfectly inelastic
In a perfectly inelastic collision the bodies stick together and move with one common velocity v: m1u1 + m2u2 = (m1 + m2)v.
This gives the largest kinetic energy loss consistent with momentum conservation.

Total energy is always conserved; it is only kinetic energy that may fall. Momentum is conserved in all collisions in an isolated system, elastic or not.

Section 4 of 9Explosions and recoil

An explosion is a collision in reverse.

1
If a stationary object breaks into two parts the total momentum afterwards must still be zero,
  • so the parts move in opposite directions with momenta of equal size: m1v1 = −m2v2.
2
The lighter part gets the greater speed.
3
Kinetic energy increases, supplied by stored chemical or elastic energy.
4
The recoil of a gun and the thrust of a rocket (exhaust gas goes back, rocket goes forward) are examples.

Section 5 of 9Method and checks

1
Draw a before and after sketch, label masses and velocities, and mark a positive direction.
2
Write the momentum conservation equation and solve for the unknown velocity.
3
If asked about elasticity, calculate total kinetic energy before and after and compare them.
4
Check that the answer is sensible: after a collision a following body cannot pass through the body in front.

Two-dimensional collisions are handled by conserving momentum separately along two perpendicular directions; the same principles apply to vector components. Use this together with impulse (4.5.4) to find the force during the impact.

Section 6 of 9Worked reasoning on energy in collisions

A useful check is to compare kinetic energy through momentum: for a body of mass m, kinetic energy Ek = p2/2m.

Worked reasoning on energy in collisions
When two bodies have equal and opposite momenta before a perfectly inelastic collision, they both stop,
  • so all of the kinetic energy is transferred to other forms.
When one body strikes an identical stationary body in an elastic head-on collision, the first stops and the second moves off with the original velocity, an exchange of momentum seen in a Newton's cradle.

A heavy body hitting a light stationary body continues forwards at slightly reduced speed, while the light body is thrown forwards at nearly twice the speed.

Worked reasoning on energy in collisions
A light body hitting a heavy stationary body rebounds with almost its original speed, and the heavy body gains only a little momentum-driven motion.
These limiting cases help you judge whether a numerical answer is reasonable.

In real laboratory experiments, light gates and air tracks reduce friction so that the external force is small, and momentum is conserved to within a few per cent. Any measured difference is attributed to friction, air resistance or measurement uncertainty.

Section 7 of 9Worked examples

  1. A 0.60 kg trolley moving at 1.5 m s−1 collides with a stationary 0.40 kg trolley. They stick together. Find their common speed and the kinetic energy lost.

  2. A 2.0 kg trolley at 3.0 m s−1 collides head-on with a 1.0 kg trolley moving at 2.0 m s−1 in the opposite direction.

  3. A 0.020 kg bullet is fired at 300 m s−1 from a rifle of mass 4.0 kg that is free to recoil. Find the recoil speed.

Section 8 of 9Key terms

TermMeaning
Conservation of momentumTotal momentum of an isolated system is constant.
Elastic collisionCollision in which total kinetic energy is conserved.
Inelastic collisionCollision in which kinetic energy is transferred to other forms.
Perfectly inelasticBodies stick together after colliding.
Isolated systemSystem with no resultant external force.
RecoilBackward motion of a body that ejects something forwards.

Section 9 of 9Exam technique

1
Saying momentum is not conserved in an inelastic collision. Only kinetic energy is lost.
2
Forgetting the sign of velocities when bodies move in opposite directions.
3
Testing elasticity by comparing total momentum, which is always conserved; compare kinetic energy instead.
4
Claiming energy is lost in a collision. It is transferred to other forms; total energy is conserved.
5
Applying momentum conservation when a large external force (such as friction on a rough floor) acts during the interaction.
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