Work, Energy and Power · Lesson 13 of 16
Collisions, Impulse and Energy Changes
“Classify collisions and choose a system for which momentum conservation is justified.”
• Distinguish elastic, inelastic and completely inelastic collisions. • Check external impulse. • Explain energy storage during contact and open-system pitfalls.
What makes a collision useful to analyse?
During a collision, interaction forces may vary rapidly and bodies can deform. Calculating every force at every instant is often unnecessary. If external impulse on the chosen system is negligible over the short collision interval, total momentum is conserved even while the internal collision force is large. This is a conditional statement about the system, direction and interval, rather than a guarantee attached to the word collision.
Gravity may be important over a pendulum swing yet have negligible impulse compared with a brief horizontal collision force. Horizontal momentum can then be conserved during contact even when vertical momentum is not. Conversely, a ball alone rebounding from a wall has a large external impulse and does not conserve its own momentum. Include the wall and Earth, or account explicitly for the impulse.
Energy classification refers to before and after
| Type | Endpoint kinetic energy of the isolated colliding system | Other condition |
|---|---|---|
| Elastic | Same before and after | Deformation energy is recovered |
| Ordinary inelastic | Lower after contact | Some energy remains as heat, deformation, sound or internal excitation |
| Completely inelastic | Usually the largest loss consistent with the same total momentum and fixed masses | Bodies share a common final velocity |
| Superelastic or explosive | Can be greater after contact | Internal stored energy is released |
All these processes obey total-energy accounting. Kinetic energy conservation is an extra condition for an elastic collision, not a consequence of momentum conservation alone. A general inelastic collision need not involve sticking. The statement “final kinetic energy is always smaller in every non-elastic event” excludes the possibility of internal chemical or other stored energy being released.
What happens while elastic bodies touch?
Two approaching elastic balls compress. As compression increases, some translational kinetic energy becomes elastic deformation energy. At maximum compression, the bodies can have the same instantaneous velocity; in their centre-of-mass frame their translational velocities can both be zero. The stored energy then accelerates them apart. Total momentum stays constant throughout an isolated collision, but translational kinetic energy generally does not stay constant during contact, even when its final value is fully recovered.
If the only interaction is described by a time-independent potential depending on separation, with no inaccessible internal excitation or dissipation, equal initial and final far-separation potential values imply an elastic scattering event. Real deformable bodies may have additional internal degrees of freedom, so a simple separation potential is an idealisation. Always state which energies the model includes.
Problem
A molecule hits a smooth stationary massive wall at 200 m/s and 30° to its normal, then rebounds at the same speed and angle. Discuss momentum and energy.
- 1.Tangential velocity is unchanged and the normal component reverses. Molecule momentum changes by magnitude 2m×200 cos30° along the wall normal.
- 2.The molecule alone therefore does not conserve momentum. The wall receives the opposite impulse; molecule plus wall plus its support can conserve total momentum when external impulse on that larger system is negligible.
- 3.The equal molecule speeds represent the ideal infinitely massive stationary-wall elastic model. For a finite freely recoiling wall, exact momentum and energy conservation require its small recoil and a corresponding small change in molecule speed.
- 4.In the massive-wall limit the wall’s recoil energy is negligible while it can still receive finite momentum. Momentum and kinetic energy are different quantities.
A leaking trolley is not a fixed-mass collision
Problem
A 300 kg trolley with 25 kg sand moves at 27 km/h on a frictionless horizontal track. Sand leaks vertically relative to the trolley at 0.05 kg/s. Find final trolley speed.
- 1.Each grain leaves carrying the trolley’s existing horizontal velocity. There is no horizontal ejection velocity relative to the trolley and no external horizontal force in the model.
- 2.The trolley receives no horizontal thrust, so its speed remains 27 km/h = 7.5 m/s. The sand leaks out in 25/0.05 = 500 s.
- 3.Do not set the momentum of the shrinking trolley-plus-remaining-sand alone constant: departing sand carries momentum across that boundary. Include all the sand as a fixed collection or account for momentum flux.
Check your understanding
Quiz
Momentum conservation for the colliding system requires
Explanation: External impulse controls total momentum change.
An inelastic collision must
Explanation: Energy changes form; sticking describes only the completely inelastic case.
During an elastic collision, translational K is
Explanation: The endpoint kinetic energy is recovered after contact.
A ball alone rebounding from a wall conserves
Explanation: The wall gives the ball an external impulse.
A trolley leaking sand with no relative horizontal ejection speed has
Explanation: There is no horizontal thrust in the stated model.
Key Takeaways
• Specify the system and test external impulse before conserving momentum. • Elasticity is an endpoint kinetic-energy condition. • Energy can pass through deformation or internal stores during contact.