First slide

Gravitaional potential energy

Question

A projectile is fired with a speed less than the escape speed. Its mechanical energy is :

Moderate

A

positive at all positions
B

negative at all positions
C

positive at all positions except the highest point where it is zero
D

none of the above

Solution

For a bounded system, total mechanical energy is negative at all positions.
$E = K + U = \frac{1}{2} m v^{2} - \frac{G M m}{R} = \frac{1}{2} m (v^{2} - {v_{e}}^{2})$ . Since $v < v_{e}$ , $E < 0$ .

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