Question

A satellite of mass m and radius r is projected with a velocity $v_0$ tangential to the surface of the earth (radius R and mass M) at the North Pole. Given r < < R). If that the satellite will go to infinity, for $R{V}_0^2=xGM$. Find x ___

Answer 1

When the centre of the satellite is at a distance y from the centre of the earth,
By conservation of energy
$\frac{1}{2}m{v}^2-\frac{GMm}{R+r}=\frac{1}{2}m{v}_1^2-\frac{GMm}{y}$
Since r << R,
$\frac{1}{2}m{v}^2-\frac{GMm}{R}=\frac{1}{2}m{v}_1^2-\frac{GMm}{y}$
Also by conservation of angular momentum at the for that distance
mvR = m \v_1\y
$v_1=\frac{vR}{y}$
$\therefore \frac{1}{2}m{v}^2-\frac{GMm}{R}=\frac{1}{2}m{v}_1^2-\frac{GMm}{y}=\frac{1}{2}\frac{m{v}^2{R}^2}{y^2}-\frac{GMm}{y}$
$\frac{1}{2}m{v}^2[1-\frac{R^2}{y^2}]=GMm[\frac{1}{R}-\frac{1}{y}]$
$\Rightarrow \frac{1}{2}m{v}^2\left[\frac{(y-R)(y+R)}{y^2}\right]=GMm\left[\frac{y-R}{Ry}\right]\Rightarrow v^2(y+R)R=2GMy$
$v^2{R}^2=y[2GM-v^{2}R]\Rightarrow y=\frac{v^2{R}^2}{2GM-v^{2}R}$
$IfR{V}^2=2GM$
$y=\infty$
$\therefore$ Satellite will escape to infinity.