Question

A $15g$ ball is shot from a spring gun whose spring has a force constant $600N{m}^{-1}$. The spring is compressed by $5cm$. The greatest possible horizontal range of the ball for this compression (Take $g=10m{s}^{-2})$

• A. $6m$
• B. $8m$
• C. $10m$
• D. $12m$

Answer 1

Answer: (C)

$10m$

Given,

$k=600N/m$

$m=15g=0.015kg$

$x=5cm=0.05m$

$g=10m/s^2$

The energy stored in the spring is given by

$E=\frac{1}{2}k{x}^2$

$E=\frac{1}{2}\times 600\times 0.05\times 0.05=0.75J$

This energy appears as kinetic energy,

$E=\frac{1}{2}m{v}^2$

$v=\sqrt{\frac{2E}{m}}$

$v=\sqrt{\frac{2\times 0.75}{0.015}}=10m/s$

Horizontal range in projectile motion,

$R=\frac{v^{2}sin2\theta }{g}$. . . . .(1)

for maximum range, $\frac{dR}{d\theta }=0$

$\frac{v^2}{g}\times 2cos2\theta =0$

$cos2\theta =0$

$2\theta =\frac{\pi }{2}$

$\theta =\frac{\pi }{4}={45}^0$

From equation (1),

$R=\frac{10\times 10sin2\times {45}^0}{10}$

$R=\frac{10\times 10sin{90}^0}{10}=10m$

The correct option is C.