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

Answer: (B)

The kinetic energy of the projectiles is maximum at the point of release. The ratio of K.E. of projectiles
$\frac{\frac{1}{2}m{u}_1^2}{\frac{1}{2}m{u}_2^2}=\frac{4}{1}$
$\frac{u_1^2}{u_2^2}=\frac{4}{1}.....\left(i\right)$
The ratio of maximum heights of projectiles
$\frac{H_1}{H_2}=\frac{4}{1}$
$\frac{\frac{u_1^{2}si{n}^2{\theta }_1}{2g}}{\frac{u_2^{2}si{n}^2{\theta }_2}{2g}}=\frac{4}{1}$
$\frac{u_1^2}{u_2^2}\times \frac{si{n}^2{\theta }_1}{si{n}^2{\theta }_2}=\frac{4}{1}$
$\frac{4}{1}\times \frac{si{n}^2{\theta }_1}{si{n}^2{\theta }_2}=\frac{4}{1}$ (from equation $\left(i\right)$)
$si{n}^2{\theta }_1=si{n}^2{\theta }_2$
${\theta }_1={\theta }_2$
Now, ratio of ranges of projectiles
$\frac{R_1}{R_2}=\frac{u_1^{2}sin2{\theta }_1}{u_2^{2}sin2{\theta }_2}$
$\frac{R_1}{R_2}=\frac{u_1^{2}sin2{\theta }_1}{u_2^{2}sin2{\theta }_1}\left(\therefore {\theta }_1={\theta }_2\right)$
$\frac{R_1}{R_2}=\frac{u_1^2}{u_2^2},\frac{R_1}{R_2}=\frac{4}{1}$