Question

• which satellite moves faster: one at 300 km altitude or one at 36,000 km altitude? explain briefly.

Explanation:

To determine which satellite moves faster, we use the formula for orbital speed \( v = \sqrt{\frac{GM}{r}} \), where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the Earth's center to the satellite (radius of Earth + altitude). A satellite at 300 km altitude has a smaller \( r \) (since \( r = R_{Earth} + h \), and \( h = 300 \) km is much smaller than \( 36,000 \) km) than the satellite at 36,000 km altitude. From the formula, orbital speed \( v \) is inversely proportional to the square root of \( r \) (for a given central mass \( M \)). So, a smaller \( r \) leads to a larger \( v \). Thus, the satellite at 300 km altitude moves faster.

Answer:

The satellite at 300 km altitude moves faster. This is because orbital speed \( v=\sqrt{\frac{GM}{r}} \) (where \( r \) is distance from Earth’s center, \( r = R_{Earth}+h \)). The 300 km altitude satellite has a smaller \( r \) (since \( h = 300 \) km \( \ll 36,000 \) km), and since \( v \) is inversely proportional to \( \sqrt{r} \) (for constant \( GM \)), a smaller \( r \) gives a larger \( v \).