Question

A mass m moving horizontally (along the x-axis) with velocity v collides and sticks to a mass of 3m moving vertically upward (along the y-axis) with velocity 2v. The final velocity of the combination is :
A. $\dfrac{3}{2}v \hat{i} +\dfrac{1}{4}v \hat{j}$
B. $\dfrac{1}{4}v \hat{i} +\dfrac{3}{2}v \hat{j}$
C. $\dfrac{1}{3}v \hat{i} +\dfrac{2}{3}v \hat{j}$
D. $\dfrac{2}{3}v \hat{i} +\dfrac{1}{3}v \hat{j}$

Answer 1

Hint: This is a question about collision of two particles. Conservation of momentum is only to be applied here. After the collision one of the particles sticks to another. This clearly implies that only one product is formed in the final state.

Formula used: Conservation of momentum:
$\vec{p_i} = \vec{p_f}$
Total initial momentum is equal to total final momentum.

Complete step by step answer:
We are given that the particle with mass m, travels with a velocity v along the x direction and another particle with mass 3m travels along the y direction. Therefore, the initial Momentum becomes:
$mv \hat{i}+ 3m. 2v \hat{j} $
Which gives
$mv \hat{i}+ 6mv \hat{j} $

Now, the two masses stick together after the collision and form a single mass of m+3m i.e., 4m. This mass moves with some final velocity say u, and then the final momentum is given as:
$4m \vec{u}$

We are required to find the final velocity so, we equate the momentum before collision to the momentum after collision, we get:
$mv \hat{i}+ 6mv \hat{j} = 4m \vec{u}$
Cancelling m on both sides and dividing both sides by 4, we get:
$\dfrac{v}{4}\hat{i}+ \dfrac{6v}{4} \hat{j} = \vec{u}$

Thus we get our required velocity as
$\dfrac{v}{4}\hat{i}+ \dfrac{3v}{2} \hat{j} $ which is option (B).

So, the correct answer is “Option B”.

Additional Information: In questions about collisions two laws are extremely important:
1. Conservation of momentum
2. Conservation of energy.
When particles collide making some angle or their directions after collision make some angle then, the sine or cosine components of the corresponding vectors should be used in the analysis.

Note: It is really important to account for x and y direction in this question by using units vectors in the initial momentum expression because we are asked to find the final vector. Also, since the two masses stick together after the collision, we only need to take one mass (or one term) in the final momentum expression.