A ball is let fall from a hight h0.There are n collisions with the earth . If the velocity of rebound after n collisions is vn and the ball rises to a hight hn , then coefficient of restitution e is given by ,
The coefficient of restitution, denoted by e, is a measure of how elastic a collision is between two objects. It is defined as the ratio of the final relative velocity of separation to the initial relative velocity of approach between the two objects.
In the case of a ball being dropped and rebounding off the ground, let's derive the expression for the coefficient of restitution using the given information.
Step 1: Determine the initial velocity of the ball just before the first collision with the ground.
The initial velocity of the ball just before the first collision can be obtained using the concept of free fall. Since the ball is dropped from a height h0, we can use the equation of motion for an object in free fall:
v₀² = u² + 2gh₀
where v₀ is the final velocity just before the first collision, u is the initial velocity (which is 0 in this case), g is the acceleration due to gravity, and h₀ is the initial height.
Since the ball is only dropping vertically downwards, the direction of the velocity just before the first collision is also downward. Thus, the initial velocity of separation (relative velocity) is -v₀.
Step 2: Determine the final velocity of the ball just after the first collision with the ground.
When the ball rebounds off the ground, the relative velocity changes direction, and we need to account for the coefficient of restitution. The magnitude of the final velocity of approach (which is also the initial velocity of separation for the nth collision) is vn. Therefore, the initial velocity of approach (relative velocity) for the nth collision is vn.
Step 3: Derive the expression for the coefficient of restitution.
Using the definition of the coefficient of restitution, we have:
e = (final relative velocity of separation) / (initial relative velocity of approach)
For the first collision, the final relative velocity of separation is -vn (since the direction changes upon rebounding), and the initial relative velocity of approach is -v₀.
By substituting these values, we get:
e = -(-vn) / (-v₀)
e = vn / v₀
For the general case of n collisions, we can apply the concept of conservation of mechanical energy to state that at each collision, the kinetic energy is reduced by a factor of e². Therefore, the final velocity of separation after n collisions, vn, can be related to the initial velocity of the ball just before the first collision, v₀, using the formula:
vn = v₀ * e^n
Similarly, the height achieved after n collisions, hn, can be related to the initial height of the ball, h₀, using the formula:
hn = h₀ * e^(2n)
Therefore, the coefficient of restitution e is given by:
e = sqrt(hn / h₀)