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Question

n elastic balls are placed at rest on a smooth horizontal plane which is circular at the ends with radius r as shown in the figure. The masses of the balls are m,m2,m22......m2n1 respectively. What is the minimum velocity which should be imparted to the first ball of mass m such that nth ball completes the vertical circle:
117849.png
  1. (34)n15gr
  2. (43)n15gr
  3. (32)n15gr
  4. (23)n15gr

A
(43)n15gr
B
(34)n15gr
C
(32)n15gr
D
(23)n15gr
Solution
Verified by Toppr

Using the expressions for velocities of the two particles in one dimensional elastic collision,

v2=2m1v1m1+m2+m1m2m1+m2v2;

Velocity of the second ball v2=0;
Let velocity of the first ball v1=v;
For collision between the first and second ball,

v2=2mvm+m2=43v;

The second ball will have a elastic collision with the third ball.
Since the ratio of consecutive masses of the balls is same, the velocity ratio of the ball will be same.

v3=43v3=(43)2v;

Continuing this for the n-th ball, we get

vn=43vn=(43)n1v;

From motion of a particle in a vertical circle, the minimum velocity required for the particle to reach the highest point on the vertical circle is (5gR)12;

vn=(5gR)12;

(5gR)12=(43)n1v;

v=(34)n1(5gR)12.
v=(34)n15gr

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