Newtonian mechanics

Elastic collisions

An elastic collision conserves both the system's total momentum and total kinetic energy.

In the ideal rigid-body model, an elastic collision does not convert total kinetic energy into heat, sound, or permanent deformation. Individual objects may still exchange kinetic energy.

m_1u_1+m_2u_2=m_1v_1+m_2v_2, \qquad \frac12m_1u_1^2+\frac12m_2u_2^2=\frac12m_1v_1^2+\frac12m_2v_2^2

Definition: Coefficient of restitution

For a one-dimensional collision, coefficient of restitution $e$ is the ratio of relative separation speed to relative approach speed. An ideal elastic collision has $e=1$ ; a perfectly inelastic collision has $e=0$ .

Vary masses and initial velocities to see both conservation equations and the resulting velocities.

One-dimensional collisions

Together with momentum conservation, kinetic-energy conservation determines the two final velocities. Equivalently, in an ideal elastic collision the relative separation speed equals the relative approach speed.

Example: Two equal-mass carts

Two identical carts: cart 1 moves right at 4 m/s and cart 2 is at rest. They collide head-on elastically. Find the final velocities.

Solution

For equal masses, the objects exchange velocities: $v_1=0$ , $v_2=4\,\mathrm{m/s}$ to the right. Both momentum and kinetic energy are unchanged.

Quick check

The two conservation equations can be combined to give a relative-velocity relation: $v_1-v_2=-(u_1-u_2)$ with a consistent sign convention. If the masses differ, they do not generally exchange velocities; that simple result applies when the masses are equal and the second object starts at rest. The elastic model is a useful approximation for billiard balls or gas molecules under idealized conditions.

If one body is much more massive, its velocity usually changes less because the same momentum change gives $Delta v=Delta p/m$ . Yet the interaction forces on the two bodies are equal and opposite by Newton’s third law, so momentum is exchanged internally. Distinguish kinetic-energy conservation in the collision from conservation of total energy in all forms. Check both equations after solving to catch sign errors.

Which quantities remain unchanged in an elastic collision of an isolated system?

Two equal masses: one moves and the other is at rest. What happens after an ideal 1D elastic collision?

References

Young, Freedman (2019). University Physics