Question

Two projectiles of same mass have their maximum kinetic energies in ratio $4:1$ and ratio of their maximum heights is also $4:1$. Then what is the ratio of their ranges?

• A. $2:1$
• B. $4:1$
• C. $8:1$
• D. $16:1$

Answer 1

The correct option is B $4:1$

Given : $m_1=m_2=m$

Let the projectiles are projected with respective velocity $u_1$ and $u_2$.

$\frac{K.E_1}{K.E_2}=\frac{4}{1}$

$\frac{\frac{1}{2}m{u}_1^2}{\frac{1}{2}m{u}_2^2}=\frac{4}{1}$ $⟹u_1=2u_2$ ..............(1)

Maximum height attained by the projectile $H=\frac{u^{2}si{n}^2\theta }{2g}$

Also it is given that $H_1:H_2=4:1$

$\therefore$ $\frac{u_1^{2}si{n}^2{\theta }_1}{u_2^{2}si{n}^2{\theta }_2}=\frac{4}{1}$

$\frac{4u_2^{2}si{n}^2{\theta }_1}{u_2^{2}si{n}^2{\theta }_2}=\frac{4}{1}$ $⟹{\theta }_1={\theta }_2$ (using (1))

Range of the projectile $R=\frac{u^{2}sin2\theta }{g}$

$\therefore$ $\frac{R_1}{R_2}=\frac{u_1^{2}sin2{\theta }_1}{u_2^{2}sin2{\theta }_2}$

$\frac{R_1}{R_2}=\frac{4u_2^{2}sin2{\theta }_2}{u_2^{2}sin2{\theta }_2}=\frac{4}{1}$