Final answer:
The gravitational potential energy of a 500 kg satellite 200 km above Earth's surface is calculated using the formula for GPE and is found to be -4.77×10¹ joules. The calculation involves the mass of the Earth, the mass of the satellite, and the total distance from the Earth's center.
Step-by-step explanation:
The gravitational potential energy (GPE) of a satellite in orbit can be calculated using the formula U = -GMe/m/r, where G is the gravitational constant (6.674×10⁻¹¹ N(m²/kg²)), Me is the mass of the Earth (5.972×10¹¹ kg), m is the mass of the satellite, and r is the distance from the center of the Earth to the satellite. Since the given satellite is 200 km above the Earth's surface, we must add the Earth's radius (6371 km) to this altitude to obtain the total distance from the center of the Earth to the satellite, which would be r = 6571 km or 6.571×10¶ meters. The mass of the satellite is given as 500 kg. Plugging these values in, we find the gravitational potential energy.
Firstly, converting units and calculating r:
6.571×10¶ m (total radius including altitude above the Earth's surface).
Using the GPE formula:
U = - (6.674×10⁻¹¹ N(m²/kg²) × 5.972×10¹¹ kg × 500 kg) / 6.571×10¶ m
U = - (19.9 × 10¹¹ N(m/kg) × 500 kg) / 6.571×10¶ m
U = -4.77×10¹ J (Gravitational Potential Energy).
Therefore, the gravitational potential energy of the satellite is -4.77×10¹ joules.