A rubber ball is released from a height of 5 m above the floor. It bounces back repeatedly, always rising to 81/100 of the height through which it falls. Find the average speed of the ball. (Take g=10 ms⁻² )

The rubber ball is released from a height of 5 meters and bounces back to 81% of the height from which it falls. We will calculate the average speed of the ball using the following steps:

1. Calculate the initial velocity upon release:

The velocity with which the ball strikes the ground is determined by the equation:

v = √(2gh)

Where: g = 10 m/s² (acceleration due to gravity) and h = 5 meters (initial height).

Substituting the values:

v = √(2 × 10 × 5) = 10.0 m/s

2. Determine the rebound height after the first bounce:

The ball rebounds to 81% of the height from which it fell:

h₁ = 0.81 × h = 0.81 × 5 = 4.05 meters

3. Calculate the time taken for the initial fall and subsequent bounces:

The time to fall from a height h is given by:

t = √(2h/g)

For the initial fall:

t₀ = √(2 × 5 / 10) = 1 second

For the first bounce:

t₁ = √(2 × 4.05 / 10) ≈ 0.9 seconds

The time for each subsequent bounce decreases by a factor of √(0.81) due to the 81% rebound height.

4. Calculate the total distance traveled:

The total distance includes the initial fall and the sum of all subsequent bounces. Using the formula for an infinite geometric series:

Total Distance = h + 2 × (h₁ + h₂ + h₃ + ...)

Using the sum of the infinite geometric series with the first term h₁ = 4.05 meters and common ratio r = 0.81, we get:

Sum = h₁ / (1 - r) = 4.05 / (1 - 0.81) = 21.3684 meters

Therefore, the total distance is:

Total Distance = 5 + 2 × 21.3684 = 47.7368 meters

5. Calculate the total time taken:

The total time is the sum of the times for the initial fall and all subsequent bounces:

Total Time = t₀ + 2 × (t₁ + t₂ + t₃ + ...)

Using the sum of the infinite geometric series with the first term t₁ = 0.9 seconds and common ratio r = √(0.81), we get:

Sum = t₁ / (1 - r) = 0.9 / (1 - 0.9) = 9 seconds

Therefore, the total time is:

Total Time = 1 + 2 × 9 = 19 seconds

6. Calculate the average speed:

The average speed is the total distance divided by the total time:

Average Speed = Total Distance / Total Time = 47.7368 / 19 ≈ 2.51 m/s

Final Answer

The average speed of the ball is approximately 2.5 m/s. Therefore, the correct option is:

b) 2.50 m/s