A superball is dropped from rest from a height of 2.0m. It bounces repeatedly from the floor, as superballs are prone to do. After each bounce the ball dissipates some energy, so eventually it comes to rest. The following pattern is observed: After the 1st bounce, the ball returns to a maximum height that is 3/4 of its initial height. After the 2nd bounce, the ball returns to a maximum height that is 3/4 of its maximum height after the 1st; After the 3rd bounce, the ball returns to a maximum height that is 3/4 of its maximum height after the second, etc. In fact, for this particular ball, the maximum height is achieved after the nth bounce is found to be 3/4 of the maximum height achieved in the previous bounce. If this pattern is repeated, how many times will the ball bounce before coming to rest, and how long the process will take? (Neglect air friction.)

To solve this problem, we need to understand the pattern in which the maximum height decreases after each bounce. Let's break down the steps to find the answer:

Step 1: Determine the relationship between consecutive maximum heights:
The problem states that the maximum height after each bounce is 3/4 of the maximum height achieved in the previous bounce. This can be represented mathematically as:

hₙ = (3/4) * hₙ₋₁

Where:
hₙ is the maximum height after the nth bounce
hₙ₋₁ is the maximum height after the (n-1)th bounce

Step 2: Determine the number of bounces until the ball comes to rest:
Since the ball comes to rest when it reaches a maximum height of zero, we can use the given pattern to figure out when that happens. Let's analyze the maximum height after each bounce:

After 1st bounce: h₁ = (3/4) * h₀
After 2nd bounce: h₂ = (3/4) * h₁ = (3/4) * (3/4) * h₀ = (3/4)² * h₀
After 3rd bounce: h₃ = (3/4) * h₂ = (3/4) * (3/4)² * h₀ = (3/4)³ * h₀
After nth bounce: hₙ = (3/4)^n * h₀

To find out when the ball comes to rest, we need to find the value of n for which hₙ equals zero. Let's solve the equation:

(3/4)^n * h₀ = 0

Since h₀ is the initial height of 2.0m, we have:

(3/4)^n * 2.0 = 0

(3/4)^n = 0

Since any non-zero number raised to the power of zero is equal to one, the only way for the equation to hold true is if (3/4)^n is equal to zero, which is not possible.

Therefore, the ball will never come to rest, as the maximum height will keep getting smaller with each bounce, but it will never reach zero.

Step 3: Determine the time it takes for the process:
Since the ball is dropped from rest, we can use the equation for free fall to determine the time it takes for the ball to reach the ground initially:

h₀ = (1/2) * g * t₀²

Where:
h₀ is the initial height (2.0m)
g is the acceleration due to gravity (approximately 9.8 m/s²)
t₀ is the time it takes for the ball to reach the ground initially

Rearranging the equation, we have:

t₀² = (2 * h₀) / g
t₀ = √((2 * h₀) / g)

Substituting the values, we get:

t₀ = √((2 * 2.0) / 9.8)

Calculating the result, we find:

t₀ ≈ 0.641s

Therefore, the initial time it takes for the ball to reach the ground is approximately 0.641 seconds.