Question

A 1000.-kg communications satellite is released from a space shuttle to initially orbit the Earth at a radius of \(7.00 \cdot 10^{6} \mathrm{~m}\). After being deployed, the satellite's rockets are fired to put it into a higher altitude orbit of radius \(5.00 \cdot 10^{7} \mathrm{~m} .\) What is the minimum mechanical energy supplied by the rockets to effect this change in orbit?

Short answer

Answer: To find the minimum mechanical energy supplied, first calculate the change in gravitational potential energy (\(\Delta U = U_2 - U_1\)) and assume the change in kinetic energy (\(\Delta K\)) is zero. Then, sum up the changes in gravitational potential energy and kinetic energy to calculate the total mechanical energy change required (\(\Delta E = \Delta U + \Delta K\)). Plug in the values from the given problem to find the minimum mechanical energy supplied by the rockets.

Step-by-step solution

Calculate Initial Gravitational Potential Energy

The gravitational potential energy (GPE) is given by the formula: $$ U = -\frac{GM_Em}{r} $$ where \(G\) is the gravitational constant (\(6.674 \times 10^{-11} N m^2 kg^{-2}\)), \(M_E\) is the mass of the Earth (\(5.972 \times 10^{24} kg\)), \(m\) is the mass of the satellite (\(1000 kg\)), and \(r\) is the distance from the center of the Earth to the satellite. Plug in the values for the initial orbit radius (\(7.00 \cdot 10^{6} m\)) and other constants to calculate the initial GPE: $$ U_1 = -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(1000)}{7.00 \cdot 10^{6}} $$

Calculate Final Gravitational Potential Energy

Using the same formula as in step 1, calculate the GPE for the final orbit with a radius of \(5.00 \cdot 10^{7} m\): $$ U_2 = -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(1000)}{5.00 \cdot 10^{7}} $$

Calculate Change in Gravitational Potential Energy

Subtract the initial GPE from the final GPE to find the change in gravitational potential energy during the orbit transition: $$ \Delta U = U_2 - U_1 $$

Calculate Change in Kinetic Energy

Since the minimum mechanical energy is required, we assume that the change in kinetic energy is zero. This is because the satellite's speed must be kept as low as possible to minimize the required energy input. So, the change in kinetic energy is: $$ \Delta K = 0 $$

Calculate the Total Mechanical Energy Change

Finally, sum up the changes in gravitational potential energy and kinetic energy to calculate the total mechanical energy change required: $$ \Delta E = \Delta U + \Delta K $$ Plug the values from steps 3 and 4 into this equation to find the minimum mechanical energy supplied by the rockets to effect the orbit change.