A projectile of mas s m is fired from the surface of earth at an angle α$$=60$$∘ from the vertical. The initial speed m of the projectile is $$u=GMR$$, 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)

Question

A projectile of mas s m is fired from the surface of earth at an angle α$$=60$$∘ from the vertical. The initial speed m of the projectile is $$u=GMR$$, 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)

Infinity Learn · Accepted Answer

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 $r_{max} = \frac{3 R}{2}$

$∴ h_{max} = r_{max} - R = \frac{R}{2} = 3200 km$

Motion of planets and satellites

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Motion of planets and satellites

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Let v be the speed of projectile at the highest point. By conservation of angular momentum and conservation of mechanical energy, we havemuRsin⁡α=mvrmax and 12mu2−GMmR=12mv2−GMmrmaxSolving these two equations, we get rmax=3R2∴ hmax=rmax −R=R2=3200km

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