The formula for finding the total distance traveled by a ball after bouncing depends on a few factors, such as the initial height from which the ball is dropped and the coefficient of restitution, which measures the elasticity of the ball and the surface it bounces on.
Assuming the ball is dropped from a certain height h and undergoes a series of bounces on a perfectly elastic surface (with no energy loss), the total distance traveled by the ball after n bounces can be calculated using the following formula:
Total distance = h + 2h + 2^2h + 2^3h + ... + 2^n-1h
This is a geometric series where each term represents the distance covered during a bounce, and the common ratio is 2 (since the ball bounces back to approximately twice its initial height).
Using the formula for the sum of a geometric series, the total distance can be simplified as:
Total distance = h * (1 + 2 + 2^2 + 2^3 + ... + 2^n-1) = h * (1 - 2^n) / (1 - 2) = h * (1 - 2^n) / (-1) = h * (2^n - 1)
So, the total distance traveled by the ball after n bounces on a perfectly elastic surface would be h times the difference between 2 raised to the power of n and 1.
It's important to note that this formula assumes ideal conditions, such as perfect elasticity and no energy loss during each bounce. In reality, various factors such as air resistance, friction, and non-ideal bounces can affect the actual distance traveled by the ball.