A tennis ball dropped from a height 30m bounces 40% of the height from which it felt on its bounce. what is the vertical distance it travels before coming to rest.
let us assume that the ball is at rest when it bounces less than 1mm.
Starting at 30,000 mm, we find that on the 12th bounce it has "stopped" bouncing.
So, with r=0.4, and noting that 12 bounces involve a round trip (up and down), we have the sum of a geometric sequence:
30 + 2*30(1-.4^12)/(1-.4) = 130m
To find the vertical distance the tennis ball travels before coming to rest, we need to calculate the consecutive heights each time it bounces until it comes to rest.
1. First, we know that the ball bounces 40% of the height from which it fell. So, after the initial drop from a height of 30m, the ball will bounce back to 40% of 30m, which is 0.4 * 30m = 12m.
2. After the first bounce, the ball will fall from a height of 12m. Again, it will bounce back to 40% of 12m, which is 0.4 * 12m = 4.8m.
3. The ball will continue to bounce, each time reaching 40% of the height from which it fell, until it comes to rest.
4. We can keep repeating the process until the height becomes negligible or insignificant.
5. We can represent the consecutive heights of the bounces in a series: 30m, 12m, 4.8m, 1.92m, and so on.
6. To calculate the total vertical distance the ball travels before coming to rest, we can sum the heights of all the bounces.
Now let's calculate the total distance:
1st bounce: 30m
2nd bounce: 30m * 0.4 = 12m
3rd bounce: 12m * 0.4 = 4.8m
4th bounce: 4.8m * 0.4 = 1.92m
...
The series formed by the heights of the bounces is a geometric progression. Summing the series can be done using the formula:
sum = a / (1 - r)
where 'a' is the first term and 'r' is the common ratio.
In this case, the first term 'a' is 30m and the common ratio 'r' is 0.4.
Summing the series, we have:
sum = 30m / (1 - 0.4) = 30m / 0.6 = 50m
Therefore, the vertical distance the tennis ball travels before coming to rest is 50m.