Step-by-step explanation:
On this page, an interesting example is presented that allows the student to obtain expressions for a general case after examining the first two or three situations. It is an exercise in geometric progressions, a common topic in elementary-level Mathematics courses.
Bounces in the horizontal plane
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When a ball bounces off a rigid board, the component of the velocity perpendicular to the board decreases in value, leaving the parallel component unchanged
v x = u x
v y = -e u y
Heights of successive bounces
Suppose a ball is dropped from an initial height h . We are going to calculate the heights of the successive bounces.
1.-First bounce
The velocity of the ball before it hits the ground is calculated by applying the principle of conservation of energy
The velocity of the ball after the collision is (in modulus) v 1 = e u 1
The ball ascends with an initial velocity v 1 , and reaches a maximum height h 1 which is calculated by applying the principle of conservation of energy
2.-Second bounce
The velocity of the ball before it hits the ground is calculated by applying the principle of conservation of energy
The velocity of the ball after the collision is v 2 = e u 2
The ball ascends with an initial velocity v 2 , and reaches a maximum height h 2 which is calculated by applying the principle of conservation of energy
3.-Bounce n
After collision n , the maximum height the ball reaches is
h n = e 2n h
Loss of energy experienced by the ball
In the first collision, the ball loses an energy
In the second collision, the ball loses an energy
In the collision n the ball loses an energy
The sum of Δ E 1 + Δ E 2 + Δ E 3 +…. Δ E n is the energy lost by the ball after n collisions. After infinite collisions the ball will have lost all its initial energy mgh . We are going to check it by adding the infinite terms of a geometric progression of ratio e 2 and whose first term is Δ E 1
Time it takes for the ball to stop.
The time it takes for the ball to reach the ground when dropped from a height h from rest is
The ball bounces and rises to a height h 1 , then falls back to the ground. The time it takes to go up and down is
The ball bounces and rises to a height h 2 , and then falls back to the ground. The time it takes to go up and down is
The total time after infinite bounces is the sum of t 0 and the terms of a geometric progression whose first term is 2 t 0 e and whose ratio is e.
If the ball is given a horizontal initial velocity v x . After infinite bounces, it moves a horiz
