Question

A missile is fired for maximum range with an initial velocity of $20m/s$. If $g=10m/s^2$, the range of the missile is

• A. $20m$
• B. $40m$
• C. $50m$
• D. $60m$

Answer 1

The correct option is B $40m$

Let Velocity is $V$ and angle of projection with horizontal is $\theta$.

$\therefore \text{Range, R}=\frac{2V^{2}sin\theta cos\theta }{g}=\frac{V^{2}sin2\theta }{g}$

$\Rightarrow \frac{dR}{d\theta }=\frac{V^2}{g}2cos2\theta =0$ $\Rightarrow 2\theta ={90}^o$ $\Rightarrow \theta ={45}^o$

$\Rightarrow \frac{d^{2}R}{d{\theta }^2}=-\frac{V^2}{g}4sin2\theta$, at $\theta ={45}^o$ $\Rightarrow \frac{d^{2}R}{d{\theta }^2}=-\frac{4V^2}{g}<0$

Hence, at $\theta ={45}^o$ , Range is maximum.

$\Rightarrow Range=\frac{V^2}{g}=\frac{{20}^2}{10}=40m$