Motion in a Straight Line

Velocity of rubber ball when it strikes to the ground, v0=√2gh0
Using the formula of coefficient of restitution,
e=‌

‌ velocity after collision ‌

‌ velocity before collision ‌

⇒‌‌e=‌

v1−0

v0−0

⇒e=‌

v1

√2gh0

⇒‌‌v1=e√2gh0...(i)
As, initial height of the ball =h0.
∴ The first height of the rebound, h1=‌

v12

2g

⇒‌‌h1=e2h0 [Using Eq. (i)]
The nth height of the ball to the rebound,
hn=‌

vn2

2g

⇒hn=e2nh0
The velocity of the ball after nth rebound, vn=env0
Now, the total distance travelled by the ball after nth rebound is
H=h0+2h1+2h2+2h3...
H=h0+2e2h0+2e4h0+2e6h0...
H=h0[1+2e2(1+e2+e4+e6...)]
Using the formula,
1+e2+e4+...=‌

1

1−e2

⇒‌‌H=h0[1+2e2(‌

1

1−e2

)]
⇒‌‌H=h0(‌

1+e2

1−e2

)
⇒‌‌H=5(‌

1+(0.81)

1−(0.81)

)‌‌(∵e2=‌

h1

h0

=‌

81

100

)
H=47.6m
Now, the total time taken by the ball to come to rest
T=t0+2t1+2t2+...
T=√‌

2h0

g

+2√‌

2h1

g

+2√‌

2h2

g

+...
T=√‌

2h0

g

[1+2e+2e2+...]
T=√‌

2h0

g

[1+2e(1+e+e2+...)]
T=19s
The average velocity,
v‌avg ‌=‌

H

T

=‌

47.6

19

=2.5m∕s
Hence, the average velocity is 2.5m/s.