Question

An artillery piece of mass \( M_1 \) fires a shell of mass \( M_2 \) horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

• A. \( \frac{M_1}{M_1 + M_2} \)
• B. \( \frac{M_2}{M_1} \)
• C. \( \frac{M_2}{M_1 + M_2} \)
• D. \( \frac{M_1}{M_2} \)

Answer 1

The Correct Option is B

Given that momentum is conserved:

\[ |p_1| = |p_2| \]

Let \( p \) denote the magnitude of the momentum. The kinetic energy (KE) of an object is given by:

\[ KE = \frac{p^2}{2M} \]

Since the momenta of the artillery and the shell are equal, the kinetic energy is inversely proportional to the mass:

\[ KE \propto \frac{1}{M} \]

Thus, the ratio of kinetic energies of the artillery (\( KE_1 \)) and the shell (\( KE_2 \)) is:

\[ \frac{KE_1}{KE_2} = \frac{\frac{p^2}{2M_1}}{\frac{p^2}{2M_2}} = \frac{M_2}{M_1} \]