Question

A particle of mass $m$ is projected from the ground with initial linear momentum $p$ (magnitude) to have maximum possible range. Its minimum kinetic energy will be

• A. $\frac{p^{2}}{2 m}$
• B. $\frac{ p ^{2}}{4 m ⁡}$
• C. $\frac{p^{2}}{m}$
• D. $\frac{ p ^{2}}{3 m ⁡}$

Answer 1

Answer: (B)

At the time of projection $\theta=45^{\circ}$ and momentum $p=m u$
$\Rightarrow u =\frac{ p }{ m }$
Minimum $KE$ will be when projectile is at the highest point.
Velocity will be is $u \cos \theta$
$\therefore K_{\min }=\frac{1}{2} m( u \cos \theta)^{2}$
$=\frac{m}{2}\left(\frac{p^{2}}{m^{2}}\right)\left(\frac{1}{2}\right)=\frac{p^{2}}{4 m }$