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

Two smooth spheres made of identical material having masses ' $m$ ' and ' $2 m$ ' undergoes an oblique impact as shown in figure. The initial velocities of the masses are also shown. The impact force is along the line joining their centres. The coefficient of restitution is $5 / 9 .$ Calculate the velocities of the masses after the impact.

Accepted Answer

Let us consider that $u_{1, t}$ and $u_{2, t}$ are the initial velocity components along the common tangent direction, and $v_{1, t}$ and $v_{2, t}$ are the final velocity components along the common tangent direction. Further, as in common tangent direction the velocities of the spheres will not change. Hence, we can write:

$v_{1, t} = u_{1, t} = 4 m s^{- 1}$ and $v_{2, t} = u_{2, t} = 8 m s^{- 1}$ where the values for the components are given in the diagram below. Here, we are substituting $θ = 53 °$ .

Further, in common normal direction we can conserve the linear momentum of the system as

$2 m \times u_{1, n} - m \times u_{2, n} = 2 m v_{1, n} + m v_{2, n}$

Putting values from figure for $u_{1, n}$ and $u_{2, n}$ , we get:

$\Rightarrow 2 m \times 3 - m \times 6 = 2 m v_{1, n} + m v_{2, n}$

or $2 v_{1, n} + v_{2, n} = 0 . . . . . (i)$

Restitution equation $e = \frac{R e l a t i v e v e l o c i t y o f s e p a r a t i o n}{R e l a t i v e v e l o c i t y o f a p p r o a c h}$

$\frac{5}{9} = \frac{v_{2, n} - v_{1, n}}{(3 + 6)}$

or $v_{2, n} - v_{1, n} = 5 . . . . . . . (i i)$

From equations. (i) and (ii),

$v_{1, n} = - \frac{5}{3} m / s$ and $v_{2, n} = \frac{10}{3} m / s$

Now as we can see in the figure that normal directions are horizontal, it means we will consider the normal components as components along the $x - a x i s$ , and similarly tangent direction is along the vertical direction, so we will consider that as the $y - a x i s$ and hence the components along tangent will be along same direction.

Hence, final velocity of ball (1), $\vec{v_{1}} = (- \frac{5}{3} \hat{i} + 4 \hat{j}) m / s$

And final velocity of ball (2), ${\vec{v}}_{2} = (\frac{10}{3} \hat{i} - 8 \hat{j}) m / s$

Important Questions on Impulse and Collision

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