A ball dropped freely from a height of 16 m.Each time it drops h metre, it rebounds 0.81 h metre. A,what is the total vertical distance travelled by the ball before it comes to rest? B,the ball takes the FF times for each fall.What is the total time before it comes to rest?

total distance

= origininal drop of 16 m + up and down of each bounce + .....
= 16 + 2(.81)(16) + 2(.81)^2 (16) + 2(.81)^3 (16) + ....

from the 2nd term on, we have an infinite GS
with a = 32 , and r = .81
so total distance
= 16 + 32/(1-.81)
= 16 + 32/(19/100)
= 16 + 3200/19
= 3504/19 m or appr 184.42 m

oops, just noticed that you wanted the total time, not the total distance.
No problem, we just have to find the time it took to fall 184.42 m

16t^2 = 3504/19
t^2 = 219/19
t = appr 3.4 sec

To find the total vertical distance traveled by the ball before it comes to rest, we need to calculate the sum of the distances for each fall and rebound.

Given:
Initial height (h1) = 16 m
Rebound factor (r) = 0.81

A) Calculate the total vertical distance traveled by the ball before it comes to rest:

Distance for the first fall (d1) = h1 = 16 m
Distance for the first rebound (d2) = r * h1 = 0.81 * 16 m = 12.96 m

The total distance traveled after the first fall and rebound is:
Total Distance (D1) = d1 + d2 = 16 m + 12.96 m = 28.96 m

This total distance becomes the updated height for the next fall:

Updated height (h2) = Total Distance (D1) = 28.96 m

Distance for the second fall (d3) = h2 = 28.96 m
Distance for the second rebound (d4) = r * h2 = 0.81 * 28.96 m = 23.4796 m

The total distance traveled after the second fall and rebound is:
Total Distance (D2) = d3 + d4 = 28.96 m + 23.4796 m = 52.4396 m

Continuing this pattern, we can calculate the distances for each subsequent fall and rebound until the ball comes to rest.

B) To calculate the total time before it comes to rest, we need the time taken for each fall and rebound.

Since we know the distance and acceleration due to gravity, we can use the equations of motion to find the time taken for each fall and rebound.

The time taken for an object to fall freely can be calculated using the formula:
Time (t) = √(2 * distance / acceleration due to gravity)

The time taken for the ball to rebound can be calculated using the formula:
Time (t) = √(2 * distance / acceleration due to gravity)

We can calculate the total time by summing up the times for each fall and rebound.

Let's calculate the distances and times step-by-step.

To find the total vertical distance traveled by the ball before it comes to rest, we can first calculate the distance traveled during each fall and rebound.

Given that the ball is dropped from a height of 16 m, we can break down the motion of the ball into consecutive falls and rebounds until it comes to rest.

Let's calculate the total vertical distance traveled by the ball:

Initially, the ball falls freely for the first time from a height of 16 m.
Distance fallen = 16 m

After the first fall, the ball rebounds.
Distance rebounded = 0.81 * distance fallen = 0.81 * 16 m

The ball again falls from the rebounded height.
Distance fallen = distance rebounded

After the second fall, the ball rebounds.
Distance rebounded = 0.81 * distance fallen

This process continues until the ball comes to rest. We can continue the calculation until the distance rebounded becomes negligible.

To find the total distance traveled by the ball, we can sum up all the distances fallen and rebounded.

Now let's calculate the total distance:

Total Distance = distance fallen (initial drop) + distance rebounded (after the first fall) + distance fallen (after the rebound) + distance rebounded (after the second fall) + ...

Total Distance = 16 m + 0.81 * 16 m + 0.81 * (0.81 * 16 m) + 0.81 * (0.81 * (0.81 * 16 m)) + ...

We can use the formula for the sum of an infinite geometric series to find the total distance traveled.

Formula for the sum of an infinite geometric series: S = a / (1 - r)

Where:
S = sum of the series
a = first term
r = common ratio

In this case, the first term (a) is 16 m, and the common ratio (r) is 0.81.

Total Distance = 16 m / (1 - 0.81)

Now, let's calculate the total vertical distance traveled by the ball.

Total Distance = 16 m / (0.19)

Total Distance = 84.21 m (approximately)

Therefore, the total vertical distance traveled by the ball before it comes to rest is approximately 84.21 m.

Now let's move on to part B: the time taken for each fall (FF) and the total time before the ball comes to rest.

Each fall and rebound of the ball takes a certain amount of time. Let's call this time FF.

The total time before the ball comes to rest will be the sum of the time taken for each fall and rebound.

Since the ball falls freely, we can use the equation for the time taken to fall freely from a height h:

Time taken to fall (T) = sqrt((2 * h) / g)

Where:
T = time taken to fall
h = height
g = acceleration due to gravity (~9.8 m/s^2)

To find the total time taken before the ball comes to rest, we need to consider both the fall and the rebound.

Total Time = time taken for each fall (FF) + time taken for each rebound (FF)

Total Time = 2 * FF

In this case, we need to calculate FF.

Using the equation for the time taken to fall freely, we have:

FF = sqrt((2 * h) / g)

FF = sqrt((2 * 16 m) / 9.8 m/s^2)

Now, let's calculate FF:

FF = sqrt(32 / 9.8)

FF ≈ 1.80 seconds (approximately)

The total time before the ball comes to rest will be twice the time taken for each fall:

Total Time = 2 * FF

Total Time = 2 * 1.80 seconds

Total Time ≈ 3.60 seconds (approximately)

Therefore, the ball takes approximately 3.60 seconds before it comes to rest.