Elastic collisions in one dimension

Section FM1-4

2 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

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FM1-4.1

Direct impact of elastic spheres. Newton's law of restitution. Loss of kinetic energy due to impact.

For a direct impact, conserve signed momentum along the line of centres when the external impulse during impact is negligible.
Newton's law gives speed of separation $=e\times$ speed of approach, with $0\le e\le1$ for the collisions in this course.
Solve the momentum and restitution equations together; check afterwards that the calculated velocities really describe separation rather than continued approach.
Momentum is conserved in an isolated impact, but kinetic energy is conserved only when $e=1$ ; calculate loss as kinetic energy before minus kinetic energy after.

Tier 1 · Easy

2 marks

ORIGINAL

Sphere $A$ approaches stationary sphere $B$ at $7\,\text{m s}^{-1}$ . After their direct impact, $A$ continues in the same direction at $2\,\text{m s}^{-1}$ . The coefficient of restitution is $0.4$ . Find the velocity of $B$ .

Tier 2 · Standard

5 marks

ORIGINAL

A $2\,\text{kg}$ sphere travelling at $6\,\text{m s}^{-1}$ strikes a stationary $3\,\text{kg}$ sphere directly. The coefficient of restitution is $1/2$ . Find both velocities after impact and the kinetic energy lost.

Tier 3 · Hard

6 marks

ORIGINAL

A $1\,\text{kg}$ sphere moving at $5\,\text{m s}^{-1}$ collides directly with a stationary $2\,\text{kg}$ sphere. The collision loses $16/3\,\text{J}$ of kinetic energy. Determine the coefficient of restitution and both velocities after impact.

FM1-4.2

Successive direct impacts of spheres and/or a sphere with a smooth plane surface.

Treat each impact separately and use the velocities immediately before that impact as the next restitution equation's approach velocities.
For direct impact with a fixed plane, the normal velocity reverses and its speed is multiplied by $e$ ; the plane's effectively infinite mass means its velocity stays zero.
After every impact, compare positions and velocities to decide which objects can meet next; an algebraic collision result alone does not prove another impact occurs.
Keep one positive direction throughout a sequence. Reversing the axis between impacts is a common source of incorrect restitution signs.

Tier 1 · Easy

2 marks

ORIGINAL

A sphere moving normally towards a fixed smooth wall at $5\,\text{m s}^{-1}$ has coefficient of restitution $0.7$ with the wall. State its velocity immediately after impact, taking motion towards the wall as positive.

Tier 2 · Standard

5 marks

ORIGINAL

Identical spheres $A$ and $B$ lie on a line, with $B$ between $A$ and a wall. Sphere $A$ moves towards stationary $B$ at $6\,\text{m s}^{-1}$ . Their coefficient of restitution is $1/2$ , and $B$ 's coefficient with the wall is $2/3$ . Find the velocities just after the first sphere-sphere impact and explain why $A$ and $B$ collide again after $B$ rebounds from the wall.

Tier 3 · Hard

7 marks

ORIGINAL

Three identical spheres $A$ , $B$ and $C$ are arranged in that order on a straight line. Initially $A$ moves towards the other two at $8\,\text{m s}^{-1}$ while $B$ and $C$ are at rest. Every impact has coefficient of restitution $1/2$ . Find the velocities after all impacts have occurred, and justify that no further collision follows.