13.3 Gravitational Potential Energy and Total Energy

Why not use the simpler expression $\Delta U = m g \left(y_{2} - y_{1}\right)$? How significant would the error be? (Recall the previous result, in the reference, that the value g at 400 km above the Earth is $8.67 \text{m}/\text{s}^{2}$.)

If we send a probe out of the solar system starting from Earth’s surface, do we only have to escape the Sun?

Assume you are in a spacecraft in orbit about the Sun at Earth’s orbit, but far away from Earth (so that it can be ignored). How could you redirect your tangential velocity to the radial direction such that you could then pass by Mars’s orbit? What would be required to change just the direction of the velocity?

It was shown that the energy required to lift a satellite into a low Earth orbit (the change in potential energy) is only a small fraction of the kinetic energy needed to keep it in orbit. Is this true for larger orbits? Is there a trend to the ratio of kinetic energy to change in potential energy as the size of the orbit increases?

Find the escape speed of a projectile from the surface of Mars.

What is the escape speed of a satellite located at the Moon’s orbit about Earth? Assume the Moon is not nearby.

An average-sized asteroid located $5.0 \times 10^{7} \text{km}$ from Earth with mass $2.0 \times 10^{13} \text{kg}$ is detected headed directly toward Earth with speed of 2.0 km/s. What will its speed be just before it hits our atmosphere? (You may ignore the size of the asteroid.)

(a) What is the change in energy of a 1000-kg payload taken from rest at the surface of Earth and placed at rest on the surface of the Moon? (b) What would be the answer if the payload were taken from the Moon’s surface to Earth? Is this a reasonable calculation of the energy needed to move a payload back and forth?