Question

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

• A. $\frac{p^2}{2m}$
• B. $\frac{p^2}{4m}$
• C. $\frac{p^2}{m}$
• D. $\frac{p^2}{6m}$

Answer 1

The correct option is B $\frac{p^2}{4m}$

Range will be maximum when angle of projection is ${45}^{\circ }$.
Let $u$ be the initial velocity with which the particle is projected.
Initial linear momentum is given by
$p=mu$
$p_{min}=mu\cos {45}^{\circ }=\frac{mu}{\sqrt{2}}=\frac{p}{\sqrt{2}}$
$K_{min}=\frac{p_{min}^2}{2m}=\frac{{\left(\frac{p}{\sqrt{2}}\right)}^2}{2m}=\frac{p^2}{4m}$