Name: Variable Mass System
Uploaded: 2019-07-19
Duration: 12 min 25 s
Description: In this video, we will look at how the laws of motion and their mathematical formulation apply to a variable mass system like a rocket. Don't miss it.

Question

A heavy ball of mass $2 M$ moving with a velocity $v_{0}$ collides elastically head-on with a cradle of three identical balls each of mass $M$ as shown in figure. Determine the velocity of each ball after collision.

Accepted Answer

Assume some spacing between the balls

After collision of $2 M$ with ball $' 1'$ applying conservation of linear momentum we have

$2 M v_{0} = 2 M u + M v_{1} \dots \dots (2)$

$u =$ Velocity of ball $2 M$ after collision and $v_{1}$ for ball $' 1'$

Coefficient of restitution will be $e = \frac{v_{1} - u}{v_{0}}$

$v_{1} = e v_{0} + u \dots \dots (2)$

Solving equation $(1)$ and $(2)$ for $e = 1$ we get

$u = (\frac{2 M - M}{2 M + M}) v_{0} = \frac{v_{0}}{3}$

$v_{1} = \frac{2 (2 M) v_{0}}{2 M + M} = \frac{4}{3} v_{0}$

Now ball $' 1'$ collides with $' 2'$ and itself comes to rest.

Let $v'_{1}$ be the velocity of ball $' 1'$ and $v'_{2}$ be the velocity of ball $' 2'$ after collision then from conservation of linear momentum we have

$m \frac{4 v_{0}}{3} = m v'_{1} + m v_{2} \frac{4 v_{0}}{3} = v'_{1} + v_{2} \dots \dots (3)$

and coefficient of restitution will be $e = 1 = \frac{v_{2} - v'_{1}}{\frac{4 v_{0}}{3}}$

$v_{2} - v'_{1} = \frac{4 v_{0}}{3} \dots \dots (4)$

Solving equation $(3)$ and $(4)$ we get

$v'_{1} = 0$ and $v_{2} = \frac{4 v_{0}}{3}$

Similarly for collision for ball $' 2'$ with ball $' 3'$ we get

$v'_{2} = 0$ and $v_{3} = \frac{4 v_{0}}{3}$

where $v'_{2}$ is velocity of ball $' 2'$ and $v_{3}$ is velocity of ball $' 3'$ after collision.

So the velocities of balls after collision is

Velocity of Ball $2 M =$ Velocity of Ball $' 3' = \frac{4 v_{0}}{3}$

Velocity of Ball $' 2' =$ Velocity of Ball $= 0$

A heavy ball of mass $2 M$ moving with a velocity $v_{0}$ collides elastically head-on with a cradle of three identical balls each of mass $M$ as shown in figure. Determine the velocity of each ball after collision.

Important Questions on Impulse and Collision

Learn and Prepare for any exam you want !

A heavy ball of mass 2M moving with a velocity v0 collides elastically head-on with a cradle of three identical balls each of mass M as shown in figure. Determine the velocity of each ball after collision.

Important Questions on Impulse and Collision

Learn and Prepare for any exam you want !

A heavy ball of mass $2 M$ moving with a velocity $v_{0}$ collides elastically head-on with a cradle of three identical balls each of mass $M$ as shown in figure. Determine the velocity of each ball after collision.