First slide

Motion of planets and satellites

Question

A projectile of mas s m is fired from the surface of earth at an angle $α = 60^{\circ}$ from the vertical. The initial speed m of the projectile is $u = \sqrt{\frac{GM}{R}}$ , where M and R are mass and radius of the earth. Find the height by the projectile in kilometer. Neglect air resistance and the rotation of earth. (Given R = 6400 km)

Moderate

Solution

Let v be the speed of projectile at the highest point. By conservation of angular momentum and conservation of mechanical energy, we have
$muRsin α = {mvr}_{max}$
$and \frac{1}{2} {mu}^{2} - \frac{GMm}{R} = \frac{1}{2} {mv}^{2} - \frac{GMm}{r_{max}}$
Solving these two equations, we get $r_{max} = \frac{3 R}{2}$
$∴ h_{max} = r_{max} - R = \frac{R}{2} = 3200 km$

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