a tennis ball is dropped from a height of 30mtrs, and it rebounds four-fifths of the distance it has fallen after each fall. find
1. the height of nth bounce
2. the total distance traveled on rebound before coming to rest.
The nth bounce will have height 30 * (4/5)^n
No idea how many bounces it will take before stopping.
But, to get the total travel after n bounces, just add 30 to twice the sum of n terms of the sequence (a round trip is up and back down)
it's a convergent geometric series, so you can find the sum
the sum, minus 30, is the total rebound distance
s = a / (1 - r) = 30 / (1 - 4/5)
To find the height of the nth bounce, we need to understand the pattern of the rebounds. Let's break down each step:
1. The ball is dropped from a height of 30 meters.
2. It rebounds four-fifths (4/5) of the distance it has fallen after each fall.
To find the height of the nth bounce, we'll use the following formula:
Height of nth bounce = (4/5) * (Height of previous bounce)
1. The first bounce starts from a height of 30 meters. Therefore, the height of the first bounce would be:
Height of first bounce = (4/5) * 30
Height of first bounce = 24 meters
2. For subsequent bounces, we need to calculate the height of each bounce using the formula mentioned earlier. Let's assume the height of the (n-1)th bounce is known. Then, the height of the nth bounce can be calculated as:
Height of nth bounce = (4/5) * (Height of (n-1)th bounce)
Now, let's find the height of the second bounce and the third bounce:
Height of second bounce = (4/5) * (Height of first bounce)
Height of second bounce = (4/5) * 24
Height of second bounce = 19.2 meters
Height of third bounce = (4/5) * (Height of second bounce)
Height of third bounce = (4/5) * 19.2
Height of third bounce = 15.36 meters
You can continue this calculation to find the height of any nth bounce.
To find the total distance traveled on rebound before coming to rest, we can sum up all the distances traveled during each individual bounce.
Total distance traveled on rebound = (Height of first bounce) + (Height of second bounce) + (Height of third bounce) + ...
Since the ball rebounds indefinitely, we can write the above expression as an infinite series:
Total distance traveled on rebound = (Height of first bounce) + (Height of first bounce) * (4/5) + (Height of first bounce) * (4/5)^2 + ...
We can use the formula for the sum of an infinite geometric series to find the sum of this infinite series. The formula is:
Sum = (First term) / (1 - Common ratio)
In this case, the first term is the height of the first bounce, which is 24 meters, and the common ratio is 4/5.
Total distance traveled on rebound = 24 / (1 - 4/5)
Total distance traveled on rebound = 24 / (1/5)
Total distance traveled on rebound = 24 * 5
Total distance traveled on rebound = 120 meters
Therefore, the height of the nth bounce can be calculated using the formula (4/5) * (Height of previous bounce), and the total distance traveled on rebound before coming to rest is 120 meters.