A ball of mass m is droped from a height h , losing x% of its energy(that it had before the respective drop) on each fall.Calculate how many times the ball will fall until it stops and how long it will take the ball to do that.(?)

(the ball bounces up and down)

1 Answer
May 22, 2018

This is what I get

Explanation:

Initial potential energy of ball =mgh=mgh, where gg is acceleration due to gravity
Energy lost in the first bounce=mghx/100=mghx100
=> Energy remaining after first bounce=mgh(1-x/100)=mgh(1x100)
Therefore, for the second drop height =h(1-x/100)=h(1x100) ......(1)

=> for each subsequent drop height would be multiplied by a factor =(1-x/100)=(1x100)

To calculate time t_1t1 taken for first drop we use of the kinematic expression

h=ut+1/2g t^2h=ut+12gt2

Assuming that the ball is dropped with initial velocity =0=0, we get

h=0xxt_1+1/2g t_1^2h=0×t1+12gt21
=>t_1=sqrt((2h)/g)t1=2hg .........(2)

From (1) we get time t_2t2 for the second drop as

t_2=sqrt((2h(1-x/100))/g)t2= 2h(1x100)g
=>t_2=t_1(1-x/100)^(1/2)t2=t1(1x100)12 ........(3)

For third drop as t_3t3

=>t_3=t_1(1-x/100)^(2/2)t3=t1(1x100)22 .......(4)

Theoretically ball will bounce oo times before it comes to stop. Total time taken by the ball to come to stop is sum of infinite series as given below

t_1(1+(1-x/100)^(1/2)+(1-x/100)^(2/2)+(1-x/100)^(3/2).....)

Now sum of infinite GP is given by the expression as

S_(oo)=a_1/(1-r)
where a_1 is the first term and r is common ratio and |r|<1

Hence we get total time taken by the ball to come to rest

T=t_1xx 1/(1-(1-x/100)^(1/2))
T=sqrt((2h)/g)xx1/(1-(1-x/100)^(1/2))