Question

Two satellites of same mass m are revolving around earth(M) in same orbit of radius r. Rotational directions of the 2 satellites are opposite, therefore they can collide. Total mechanical energy of the system (both satellites and earth) is (m<<M) __?
1)-GMm/r
2)-2GMm/r
3)-GMm/2r

Answer 1

Answer on Question #72147-Physics-Classical Mechanics

Two satellites of same mass $m$ are revolving around earth(M) in the same orbit of radius $r$ . Rotational directions of the 2 satellites are opposite, therefore they can collide. Total mechanical energy of the system (both satellites and earth) is $(m << M)$ ?

1)-GMm/r

2)-2GMm/r

3)-GMm/2r

Solution

Total mechanical energy of satellite A is

$$E _ {A} = E _ {p o t} + E _ {k i n}$$ $$E _ {A} = - \frac {G M m}{r} + \frac {1}{2} m v ^ {2} = - \frac {G M m}{2 r}$$

Satellites are identical, so

$$E _ {B} = E _ {A} = - \frac {G M m}{2 r}$$

Total mechanical energy of the system (both satellites and earth) is

$$E = E _ {B} + E _ {A} = - \frac {G M m}{2 r} - \frac {G M m}{2 r} = - \frac {G M m}{r}$$

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