Question

A particle of mass $1$ g moving with a velocity $\overrightarrow{u_2}=(3\hat{i}-2\hat{j})$ $m{s}^{-1}$ experiences a perfectly inelastic collision with another particle of mass $2$ g and velocity $\overrightarrow{u_2}=(4\hat{j}-6\hat{k})m{s}^{-1}$. The velocity of the combined particle is?

• A. $\hat{i}+2\hat{j}-4\hat{k}$
• B. $\hat{i}-2\hat{j}+4\hat{k}$
• C. $\hat{i}-2\hat{j}-4\hat{k}$
• D. $\hat{i}+3.33\hat{j}+4\hat{k}$

Answer 1

The correct option is A $\hat{i}+2\hat{j}-4\hat{k}$

In perfectly inelastic collision, the two colliding particle sticks together after the collision and moves with a constant velocity. the linear momentum is always conserved for perfectly inelastic collision.

so, the total linear momentum of the given system before collision is,

$P_b=\left[1gm\right(3\hat{i}-2\hat{j}\left)m/s\right]+\left[2gm\right(4\hat{j}-6\hat{k}\left)m/s\right]$

$P_b=3\hat{i}+6\hat{j}-12\hat{k}$

let the velocity of the particle after collision is $x\hat{i}+y\hat{j}+z\hat{k}$

so, the linear momentum after collision is,

$P_a=3gm(x\hat{i}+y\hat{j}+z\hat{k})$

$P_a=(3x\hat{i}+3y\hat{j}+3z\hat{k})$

from conservation of linear momentum,

$P_a=P_b$

$3\hat{i}+6\hat{j}-12\hat{k}=(3x\hat{i}+3y\hat{j}+3z\hat{k})$

comparing the $\hat{i}$, $\hat{j}$, $\hat{k}$ terms we will get the velocity of the combined particle.

$x=1$

$y=2$

$z=-4$

so, the velocity of the combined particle is,

$x\hat{i}+y\hat{j}+z\hat{k}$

$\hat{i}+2\hat{j}-4\hat{k}$