Question

An object is thrown vertically upwards from the surface of the earth, with a speed(VO). If its speed at a height (h) is(V),then show that

V0square - Vsquare=2GM/Rsquare(h/1+h/r) =2gh/(1+h/r)

Answer 1

An object is thrown vertically upwards from the surface of the earth, with a speed (VO). If its speed at a height (h) is (V), then show that

V0square - Vsquare=2GM/Rsquare(h/1+h/r) = 2gh/(1+h/r)

Solution

Using the energy conservation law ($R_{earth}$ is the radius of Earth, $M$ is the mass of Earth) we obtain

$$\frac{m v_0^2}{2} - \frac{m v^2}{2} = \frac{G m M}{R_{earth}} - \frac{G m M}{R_{earth} + h} \Rightarrow$$ $$v_0^2 - v^2 = \frac{2 G M h}{R_{earth} (R_{earth} + h)}$$

We have

$$\frac{G M}{R_{earth}^2} = g \Rightarrow$$ $$v_0^2 - v^2 = \frac{2 g}{(1 + \frac{h}{R_{earth}})}$$

Answer

$$v_0^2 - v^2 = \frac{2 g}{(1 + \frac{h}{R_{earth}})}$$