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Projectile Motion at $\frac {g}{6}$

What minimum kinetic energy should be imparted to a projectile of mass 12 g, launched from point on the Moon's surface, so that it hits a target P, where OP=200m? The height of the target above the lunar surface is 50m. Take the acceleration due to gravity on the Moon to be one-sixth of that on Earth, with $g_{Earth} = 9.8 m /{s^2}$.

Solution

Using the typical symbols associated with projectile motion, we have,

$x = u\cos \theta .t,\cos \theta  = \frac{x}{{ut}}$

and $y = u\sin \theta .t - \frac{1}{2}g{t^2},\sin \theta  = \frac{{y + \frac{1}{2}g{t^2}}}{{ut}}$

${\sin ^2}\theta  + {\cos ^2}\theta  = 1 = \frac{{{x^2}}}{{{u^2}{t^2}}} + \frac{{{{\left( {y + \frac{1}{2}g{t^2}} \right)}^2}}}{{{u^2}{t^2}}}$

$\Rightarrow {u^2} = \frac{{{x^2}}}{{{t^2}}} + \frac{{{y^2} + yg{t^2} + \frac{1}{4}{g^2}{t^4}}}{{{t^2}}}$

$ \Rightarrow {u^2} = \frac{{{x^2}}}{{{t^2}}} + \frac{{{y^2}}}{{{t^2}}} + yg + \frac{1}{4}{g^2}{t^2}$

$ \Rightarrow {u^2} = \frac{{{r^2}}}{{{t^2}}} + \frac{{{g^2}{t^2}}}{4} + yg$, $x^2+y^2=r^2$

Using $AM \ge GM$,

$\frac{{\frac{{{r^2}}}{{{t^2}}} + \frac{{{g^2}{t^2}}}{4}}}{2} \ge \sqrt {\frac{{{r^2}}}{{{t^2}}} \times \frac{{{g^2}{t^2}}}{4}} $

$\Rightarrow \frac{{{r^2}}}{{{t^2}}} + \frac{{{g^2}{t^2}}}{4} \ge gr$

Hence, ${u^2} \ge gr + yg$

$\Rightarrow {u^2} \ge g\left( {y + r} \right)$

$ \Rightarrow {u_{\min }} = \sqrt {g\left( {y + r} \right)} $ Or $K{E_{\min }} = \frac{1}{2}mg(y + r)$

This are the general formulae, in case of Moon we need to use g as $\frac {g_{Earth}}{6}$.

$K{E_{\min }} = \frac{1}{2} \times 12 \times {10^{ - 3}} \times \frac{{9.8}}{6}(50 + 200)$

$K{E_{\min }} = {10^{ - 3}} \times 9.8 \times 250 = 2.45J$

The minimum launch speed is approximately 20.2 m/s20.2\,\text{m/s}. Although a launcher designed for a scientific experiment could be made to impart this speed, many commercially available guns have considerably higher muzzle velocities. Even with a higher muzzle velocity, the target can still be hit by choosing a suitable angle of projection above the horizontal. In the ideal projectile-motion model, for every speed greater than the minimum, there are two possible launch angles that allow the projectile to reach the target. At the minimum speed, the two angles coincide.