Question

An object of mass$3m$ initially at rest splits into three equal fragments, Two fragments have velocities $v\hat{i}$ and $v\hat{j}$. What is the velocity of the third fragment?

Answer 1

Step 1: Given data

1. Mass of the object $3m$
2. Two fragments have velocities $v\hat{i}$ and $v\hat{j}$.

Step 2: Formula used

Momentum$=mv$ where $m$ is the mass and $v$ is the velocity

Step 3: Find the velocity of the third fragment

The initial momentum of $3m$ is $=0$

This mass splits into three fragments of equal masses m.

Let the velocity of the third mass be $\overrightarrow{V}$

Final momentum of the system $=m\overrightarrow{V}+mv\hat{i}+mv\hat{j}$

According to the law of conservation of momentum, for a collision between two bodies.

Total momentum before collision $=$Total momentum after the collision.

Therefore by the law of conservation of linear momentum

$$\begin{gathered} \Rightarrow m\overrightarrow{V}+mv\hat{i}+mv\hat{j}=0 \\ \Rightarrow m\overrightarrow{V}=-m\left(v\hat{i}+v\hat{j}\right) \\ \Rightarrow \overrightarrow{V}=-\left(v\hat{i}+v\hat{j}\right) \end{gathered}$$

Step 4: Find the magnitude of the velocity of the third fragment

The velocity of the third fragment is $\overrightarrow{V}=-\left(v\hat{i}+v\hat{j}\right)$

Therefore the magnitude of the velocity is

$$\begin{gathered} \Rightarrow \left|\overrightarrow{V}\right|=\sqrt{{\left(-v\right)}^2+{\left(-v\right)}^2} \\ =\sqrt{2v^2} \\ =v\sqrt{2} \end{gathered}$$

Hence the velocity of the third fragment is $-\left(v\hat{i}+v\hat{j}\right)$ and the magnitude is $v\sqrt{2}$