Question

An object, initially at rest, explodes in three fragments. The momentum of two pieces is \(-3p\hat{i}\) and \(-4p\hat{j}\) where \(p\) is a positive number. The momentum of the third piece
\((i)\) will have magnitude \(5p\)
\((ii)\) will make an angle \(\tan^{-1}(4/3)\) with the \(x\)-axis
\((iii)\) will make an angle \(\tan^{-1}(3/4)\) with the \(x\)-axis
\((iv)\) will have magnitude \(7p\).

Answer 1

Draw a diagram of given situation.

Find the momentum of the third piece.
Let the momentum of the third piece be \(p_x\hat{i}+p_y\hat{j}\).

By the momentum conservation,
\(0=-3p\hat{i}-4p\hat{j}+p_x\hat{i}+p_y\hat{j}\)

\(=(-3p+p_x)\hat{i}+(-4p+p_y)\hat{j}\)

\(-3p+p_x=0\Rightarrow p_x=3p\)

\(-4p+p_y=0\Rightarrow p_y=4p\)

The momentum of the third piece.
\(\left | \vec{P} \right |=\sqrt{(3p)^2+(4p)^2}=5p\)

\(\tan \theta =\dfrac{4p}{3p}\Rightarrow \theta =\tan^{-1}\left ( \dfrac{4}{3} \right )\)
Final Answer: (B)