Question

A mass $M$ moving with velocity $v$ along $x$-axis collides and sticks to another mass $2M$ which is moving along $y$-axis with velocity $3v$. After collision, the velocity of the combination is

• A. $\dfrac{v}{3}\hat{i} - 2v\hat{j}$
• B. $\dfrac{2v}{3}\hat{i} + \hat{j}$
• C. $v\hat{i} + \dfrac{v}{3}\hat{j}$
• D. $\dfrac{v}{3}\hat{i} + 2v\hat{j}$

Hint:

In inelastic collisions, conserve momentum separately along each axis.

Answer 1

The Correct Option is D

Step 1: Apply law of conservation of momentum.
Since the bodies stick together, the collision is perfectly inelastic, but momentum is conserved.
Step 2: Momentum along $x$-axis.
Initial momentum along $x$-axis: \[ p_x = Mv \] Step 3: Momentum along $y$-axis.
Initial momentum along $y$-axis: \[ p_y = 2M \times 3v = 6Mv \] Step 4: Total mass after collision.
\[ M_{\text{total}} = M + 2M = 3M \] Step 5: Final velocity components.
\[ v_x = \dfrac{Mv}{3M} = \dfrac{v}{3} \] \[ v_y = \dfrac{6Mv}{3M} = 2v \] Step 6: Final velocity vector.
\[ \vec{v} = \dfrac{v}{3}\hat{i} + 2v\hat{j} \]