Question

Challenge Problem: Momentum and Kinetic Energy An explosion causes an object, initially at rest, to break into two pieces. One of these pieces has twice as much kinetic energy as the other. What is the ratio of their masses? Which one has the larger mass? Is it the one with the larger kinetic energy, or the one with the smaller kinetic energy?

Answer 1

Before explosion the object is at rest, after explosion the two masses $m_1,m_2$ move with different velocities $v_1,v_2$ respectively.

Conserving linear momentum of two masses, $0=m_1v_1+m_2v_2$ (or) $p_1+p_2=0$  $\Rightarrow p_1=-p_2$ ---(1)

Kinetic energy of mass $m_1$ is $K_1=\frac{p_1^2}{2m_1}$   $\Rightarrow m_1=\frac{p_1^2}{K_1}$

Kinetic energy of mass $m_2$ is $K_2=\frac{p_2^2}{2m_2}$ , using (1) $K_2=\frac{p_1^2}{2m_2}$ $\Rightarrow m_2=\frac{p_1^2}{K_2}$

The mass $m_1$ has twice as much kinetic energy as the other mass $m_2$.. That is $K_1=2K_2$

Hence mass $m_2$ can be written as $m_2=\frac{p_1^2}{K_1/2}=2\frac{p_1^2}{K_1}=2m_1$

That is mass $m_2$ is double the mass $m_1$.

$\frac{m_1}{m_2}=\frac{1}{2}$

Mass with smaller kinetic energy has larger mass.