Question

find the minimum energy required to launch a satellite of mass m from surface of earth of mass M​

Answer 1

Answer:

Minimum energy required to launch a satellite

= $\frac{GMm}{r}$

Explanation:

In the question, we find the minimum energy required to launch a satellite of mass m from the surface of earth of mass M

We know the formula of kinetic energy and potential energy

The kinetic energy = $\frac{GMm}{2r}$

The gravitational potential energy = $\frac{GMm}{r}$

Now, required minimum energy  to launch a satellite

= Kinetic energy - potential energy

= $\frac{GMm}{2r}-\frac{GMm}{r}$

= $\frac{GMm}{r}$

Answer 2

Given:

Mass of satellite =m

Mass of Earth = M

Radius of orbit = 2R

Formula Used:

Mechanical energy is conserved. So, total mechanical energy at the surface is equal to that in the orbit.

Kinetic energy = $\frac{1}{2}mv^2$

Centripetal force $=\frac{mv^2}{r}$

Calculations:

$-\frac{GMm}{R}+K_o=-\frac{GMm}{2R+R}+\frac{1}{2}mv_o^2$

Orbital velocity is$v_o=\sqrt{\frac{GM}{3R}}$

Substitute and find value of $K_o$:

$K_o=\frac{5}{6} \frac {GMm}{R}$

Learn more about: Gravitational potential energy and orbital velocity

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