A ball is dropped from a height h above a tile floor and...

A ball is dropped from a height $h$ above a tile floor and rebounds to a height of $0.65 h$. Find the coefficient of restitution between ball and floor. Assign floor quantities the subscript 1 , and ball quantities the subscript $2 .$ The initial and final velocities of the floor, $u_{1}$ and $v_{1}$, are zero. Therefore, $$ e=\frac{v_{2}-v_{1}}{u_{1}-u_{2}}=-\frac{v_{2}}{u_{2}} $$ Since we know both the drop and rebound heights ( $h$ and $0.65 h$ ), we can write equations for the interchange of $\mathrm{PE}_{\mathrm{G}}$ and $\mathrm{KE}$ before and after the impact $\sqrt{665}=0.8$ Notice that the coefficient of restitution equals the square root of the final rebound height over the initial drop height.

A ball is dropped from a height $h$ above a tile floor and rebounds to a height of $0.65 h$. Find the coefficient of restitution between ball and floor.
Assign floor quantities the subscript 1 , and ball quantities the subscript $2 .$ The initial and final velocities of the floor, $u_{1}$ and $v_{1}$, are zero. Therefore,
$$
e=\frac{v_{2}-v_{1}}{u_{1}-u_{2}}=-\frac{v_{2}}{u_{2}}
$$
Since we know both the drop and rebound heights ( $h$ and $0.65 h$ ), we can write equations for the interchange of $\mathrm{PE}_{\mathrm{G}}$ and $\mathrm{KE}$ before and after the impact
$\sqrt{665}=0.8$
Notice that the coefficient of restitution equals the square root of the final rebound height over the initial drop height.

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Key Concepts

Coefficient of Restitution

The coefficient of restitution quantifies the elasticity of a collision by measuring the ratio of the relative speed after the collision to the relative speed before the collision along the line of impact. It indicates how much kinetic energy is conserved in the collision, with values ranging from 0 for a perfectly inelastic collision to 1 for a perfectly elastic collision.

Energy Conservation in Collisions

This concept involves the transformation between gravitational potential energy and kinetic energy. When an object is dropped from a height, its potential energy converts into kinetic energy as it falls, and upon impact, part of that kinetic energy is restored into potential energy during the rebound. This relationship allows the determination of speeds or heights using energy conservation principles.

Free Fall Kinematics

Free fall kinematics describes the motion of an object under the influence of gravity alone, without any air resistance. This includes using equations that relate an object’s height, gravitational acceleration, and velocity to analyze its motion when dropped and its subsequent rebound.

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