Why does a ball not bounce on a cushion

To study this problem one should first start with a simpler case of a ball colliding with another ball which is stationary or moving. Simplify further that the motion is one dimensional along a straight line. Impact takes place during a very short time that can be divided into two parts — deformation period and restoration period.

During the deformation period the area of contact increases from zero to a maximum value. Both the bodies suffer a momentary deformation. During the restoration period they recover their shape partly or fully.

Knowing the mass and initial velocities of each ball we can calculate the velocities after the impact by applying the principle of conservation of momentum (mass multiplied by velocity). But to do that we also need additional information on how far the contacting bodies recovered from deformation. This is expressed in a simple lumped fashion as impact coefficient (or coefficient of restitution), e. This is the ratio of relative velocity after impact to relative velocity before impact. If e is unity it means complete elastic recovery and if e is zero it is complete plastic deformation.

For a steel ball bouncing on a hard surface e is close to unity. If e is less than unity it represents some permanent deformation and it also manifests in the form of loss of kinetic energy of the system. This lost kinetic energy is converted into permanent deformation with the deformation waves dissipated as heat and also as sound.

The cushion is a good energy absorber with e close to zero. It deforms due to the ball impact and will not quickly regain its shape to push back the ball. Hence the ball will not bounce. A ball made up of cloth will not bounce on a hard floor either.