a tennis ball dropped from a height of 30m bounces 40% of the height from which it fell on each bounce. what is the vertical distance it travels before coming to rest?

first bounce = 30 m

2nd bounce = 2(.4)(30) = 60(.4) , up and down
3rd bounce = 2(.4)^2 (30)= 60(.4)^2
..
nth bounce = 60(.4)^(n-1)

total distance
= 30 + (60(.4) + 60(.4^2) + 60(.4^3) + .... )

So consider it to be an infinite geometric series starting with the second bounce
a = 60(.4)
r = .4
sum ∞ = a/(1-r)
=60(.4)/.6 = 40
but we can't forget the initial drop from 30 m

so total distance = 70 m

A vacuum pump remove 25% to the air from a scaled jar on each strike of its piston.The jar contains one litre of air before the pump starts.write a sequence to the amount of air remaining inside the jar after each stroke of the piston.will there ever be no air inside the jar?.

To solve this problem, we can find the total vertical distance traveled by the tennis ball by summing up the distances for each bounce.

Given:
Initial height (h) = 30m
Bounce factor = 40% = 0.4

Step 1: Calculate the distance traveled in the first bounce:
Distance of the first bounce = h * bounce factor
Distance of the first bounce = 30m * 0.4 = 12m

Step 2: Calculate the distance traveled in the second bounce:
Distance of the second bounce = (distance of the first bounce) * bounce factor
Distance of the second bounce = 12m * 0.4 = 4.8m

Step 3: Continue calculating the distances for each subsequent bounce until the ball comes to rest:
Distance of the third bounce = (distance of the second bounce) * bounce factor
Distance of the fourth bounce = (distance of the third bounce) * bounce factor
... and so on.

Since the ball comes to rest when the distance traveled becomes negligible, we can stop calculating when the distance becomes less than, say, 0.1m.

Now, let's calculate the distances for each bounce until the ball comes to rest:

Distance of the third bounce = 4.8m * 0.4 = 1.92m
Distance of the fourth bounce = 1.92m * 0.4 = 0.768m
Distance of the fifth bounce = 0.768m * 0.4 = 0.3072m
Distance of the sixth bounce = 0.3072m * 0.4 = 0.12288m
Distance of the seventh bounce = 0.12288m * 0.4 = 0.049152m
Distance of the eighth bounce = 0.049152m * 0.4 = 0.0196608m

Continuing this pattern, the distances become negligible and can be ignored.

Step 4: Sum up all the distances traveled by the ball to find the total vertical distance covered:
Total distance = (distance of the first bounce) + (distance of the second bounce) + (distance of the third bounce) + ...

Total distance = 12m + 4.8m + 1.92m + 0.768m + 0.3072m + 0.12288m + 0.049152m + 0.0196608m + ...

To get an approximate answer, we can round off the distances after the eighth bounce to 4 decimal places:

Total distance ≈ 12m + 4.8m + 1.92m + 0.768m + 0.3072m + 0.1229m + 0.0492m + 0.0197m = 20.9890m

Therefore, the tennis ball covers a vertical distance of approximately 20.9890m before coming to rest.

To determine the vertical distance the tennis ball travels before coming to rest, we need to calculate the total distance traveled by summing up the distances of each bounce.

Let's break down the problem step by step:

1. Determine the height of the first bounce:
- The tennis ball is dropped from a height of 30m, so the first bounce is equal to 40% of 30m: 0.4 * 30m = 12m.

2. Calculate the total distance traveled during the first bounce:
- The ball travels upward and then downward, covering a total distance twice the height of the bounce: 2 * 12m = 24m.

3. Calculate the height of the second bounce:
- The second bounce is 40% of the previous bounce's height: 0.4 * 12m = 4.8m.

4. Calculate the total distance traveled during the second bounce:
- Similar to the first bounce, the ball travels upward and then downward, so the total distance covered is 2 * 4.8m = 9.6m.

5. Continue this process until the height of the bounce is negligible (close to zero).

To find the total distance traveled before coming to rest, we need to add up all the distances from each bounce.

Total distance = 24m + 9.6m + ... (continuing with each bounce until height becomes negligible)

It's worth noting that in this problem, the heights of subsequent bounces will reduce by 40% with each iteration, forming a geometric progression. To determine the total distance covered, we can use the formula for the sum of an infinite geometric series:

Total distance = height of first bounce + height of second bounce + height of third bounce + ...

To find the sum, we use the formula:
Sum = a / (1 - r)

where:
a is the first term or height of the first bounce, which is 24m (as calculated earlier), and
r is the common ratio, which is 0.4.

Substituting the values into the formula:
Total distance = 24m / (1 - 0.4)

Simplifying the equation further:
Total distance = 24m / 0.6

Calculating the value:
Total distance = 40m

Therefore, the tennis ball travels a total vertical distance of 40 meters before coming to rest.