Momentum

Compare collisions with the same initial and final momentum. The required impulse is then unchanged; the design alters how that impulse is delivered over time. A smaller average force does not mean the momentum change is smaller, nor does it require the actual force to remain constant during the collision.

Occupant restraints and vehicle deformation must work together for the stopping-time argument to apply to the occupant. In an exam explanation, identify the relevant object and connect its momentum change to the time and average force. Avoid the vague phrase absorbs impact without explaining a physical quantity, and do not treat a simplified average-force calculation as a complete real-world safety assessment.

The word system matters. A ball striking a wall changes momentum, so momentum is not conserved for the ball considered alone. If the wall and Earth are included, the interaction transfers momentum to that larger system. The Earth's speed change is tiny because its mass is enormous, rather than because the exchanged momentum has ceased to exist.

For two trolleys on a nearly level low-friction track, collision forces can dominate over the short contact time, making external impulse negligible. Over a much longer interval, resistance may no longer be negligible. State the time interval and identify which forces are external instead of applying conservation automatically to every collection of moving objects.

The subtraction is final momentum minus initial momentum, not simply the difference between speed magnitudes. A change from rightward motion to leftward motion can therefore have a large magnitude even when the two speeds are similar. The sign of the impulse gives the direction of the momentum change in your chosen convention.

A force–time graph gives the same information through its signed area. A rectangular area applies to a constant force; a triangular pulse uses half the base times height. If positive and negative forces occur during an interval, their signed impulses can partly cancel. The average force is the net impulse divided by the full interval, not necessarily the largest force shown.

Choose a direction before substituting. A stationary object contributes zero initial momentum but still contributes mass after joining the other object. Forgetting that added mass predicts a final speed that is too large. If both objects are initially moving, keep both initial momentum terms even when their directions differ.

Do not average the velocities unless the masses happen to be equal. Momentum weights each velocity by mass, so a heavier object's motion contributes more strongly to the final result. Check your answer by multiplying the combined mass by the calculated final velocity and confirming that it reproduces the signed total momentum you started with.

Momentum is a vector and depends on signed velocity, whereas kinetic energy depends on speed squared. You cannot infer that one is unchanged just because the other is unchanged. A collision described as objects sticking together requires a momentum calculation first; assuming their total kinetic energy also stays constant would usually give an incompatible answer.

To check energy, calculate each object's kinetic energy before and after using the velocities obtained from the momentum model. Add those positive energies, then compare the totals. The decrease identifies energy transferred into other forms within the wider system. An elastic collision is the special case where both total momentum and total kinetic energy are conserved under the isolated-system assumptions.

Suppose right is positive. A ball initially moving right has positive momentum; if it rebounds left, its final momentum is negative. Subtracting the positive initial value from the negative final value gives a change whose magnitude is the sum of the two momentum magnitudes, not their difference.

This is a directional argument, not a claim that rebound always produces a greater force in every real collision. The contact time may change too, so compare both Δp and Δt before comparing average force. Keep separate the force on the ball and the opposite force it exerts on the wall; their directions belong to different objects.