Question

A bullet of mass $m = 50\;{\rm{gm}}$ strikes a sand bag of mass
$M = 5\;{\rm{kg}}$ hanging from a fixed point with a horizontal velocity ${\vec v_p}$. If a bullet sticks to the sandbag, what fraction of K.E is lost during impact?

Answer 1

The above problem is resolved by using the concepts and applications of the kinetic energy, along with the principle of the conservation of momentum. When the two objects coming towards each other suddenly collide, the net momentum will remain conserved. This concept is being applied to the given problem.

Complete step by step answer:
The mass of the bullet is $m = 50\;{\rm{gm}}$.
The mass of the sandbag is $M = 5\;{\rm{kg}}$.
As, the bullet strikes the sand bag with the magnitude of horizontal velocity ${\vec v_p}$.
Then the expression for the linear momentum just before the impact is,
${p_1} = m{v_p}$
Let the combined speed of the bullet and the sand bag be ${v_1}$.
Then momentum of the bullet and sandbag system is,
${p_2} = \left( {m + M} \right){v_1}$
It is observed that there is no net horizontal force acting on the bullet and the sandbag system. Then the net momentum before and after collision will remain conserved.

{p_1} = {p_2}\\\ m{v_p} = \left( {m + M} \right){v_1}\\\ {v_1} = \left( {\dfrac{m}{{m + M}}} \right){v_p} \end{array}$$ The mass of the bullet is much lesser than the mass of the sand bag. Then the above equation is written as, $$\begin{array}{l} {v_1} = \left( {\dfrac{m}{{m + M}}} \right){v_p}\\\ {v_1} = \left( {\dfrac{m}{M}} \right){v_p} \end{array}$$……(1) The fraction of kinetic energy lost during the impact is, $$E = {\left( {\dfrac{{{v_1}}}{{{v_p}}}} \right)^2}$$..…(2) On using the values of equation 1 and 2 as, $$\begin{array}{l} E = {\left( {\dfrac{m}{M}} \right)^2}\\\ E = {\left( {\dfrac{{50\;{\rm{gm}} \times \dfrac{{1\;{\rm{kg}}}}{{1000\;{\rm{gm}}}}}}{{5\;{\rm{kg}}}}} \right)^2}\\\ E = {10^{ - 4}} \end{array}$$ Therefore, the fraction of kinetic energy lost is of $${10^{ - 4}}$$. **Note:** Try to understand the concepts and application of momentum conservation fundamentals and the fraction of kinetic energy lost during the collision. The kinetic energy will show some variation when the collision will occur at a time instant. Moreover, the significant relation for a fraction of kinetic energy is obtained after substituting the values.