The ball is dropped from a height of 42 m . The coefficient of restitution,
e, is 0.4 . This number tells us how much energy the ball keeps after bouncing.
At first, the ball is dropped. So, its starting speed
(u) is 0 .
Height after first bounce:
The ball comes back up to a height equal to
e2 times the height it fell from. So:
‌h1=(0.4)2×42‌h1=0.16×42=6.72mHeight after later bounces:
Each time the ball bounces, it goes up to
e2 times the previous bounce's height.
h2=(0.4)2×6.72=0.16×6.72Total Distance:
When the ball is dropped, it first travels 42 m down.
After that, on every bounce, it goes up and then comes down the same height. So, for all the bounces, we call the sum of all the heights
h1,h2,h3,....
Total distance
= distance of first fall
+2× (sum of all bounce heights)
=42+2×[h1+h2+h3+...]
Summing the Series:
All the bounce heights make a series called a geometric progression (GP). The first term
a=42 and the common ratio
r=0.16.
The sum of an infinite GP is given by:
‌‌ Sum ‌=‌=‌‌‌ Sum ‌=‌=50Final Calculation:
Now, total distance is:
‌‌ Total distance ‌=42+2×(0.4)2×50‌=42+2×0.16×50‌=42+16=58m