show that potential energy remains constant throughout the distance.
Answers
Answer:
Gravitational potential energy at large distances is directly proportional to the masses and inversely proportional to the distance between them. The gravitational potential energy increases as rrr increases.
How to apply conservation laws to orbits
Although the Earth orbits the Sun, it does not go around in a perfect circle, but rather takes an elliptical path (Figure 1).
Figure 1: An elliptical path of a planet around the Sun. When the planet is closest to the Sun, speed vvv and kinetic energy are the highest, and gravitational potential energy is the lowest. When the planet moves farther away, the speed and kinetic energy decrease, and the gravitational potential energy increases. At all points in the orbit, angular momentum and energy are conserved.
This means that the Earth’s distance from Sun rrr varies throughout the orbit. There is no net external force or torque acting on the Sun-planet system, and the only force is gravity between the Sun and planet. Therefore, angular momentum and energy remain constant. However, the gravitational potential energy does change, because it depends on distance. As a result, kinetic energy also changes throughout an orbit, resulting in a higher speed when a planet is closer to the Sun.
When dealing with gravitational potential energy over large distances, we typically make a choice for the location of our zero point of gravitational potential energy at a distance rrr of infinity. This makes all values of the gravitational potential energy negative.
[Why do we choose the zero point at infinity?]
rrrr5\cdot 10^7 \,\text m
5, dot, 10, start superscript, 7, end superscript, start text, m, end text1 \%1, percent
If we make our zero of potential energy at infinity, then the gravitational potential energy as a function of rrr is:
U_G = -\dfrac{G m_1 m_2}{r}U
G
=−
r
Gm
1
m
2
U, start subscript, G, end subscript, equals, minus, start fraction, G, m, start subscript, 1, end subscript, m, start subscript, 2, end subscript, divided by, r, end fraction
For example, imagine we are landing on a planet. As we come closer to the planet, the radial distance between us and the planet decreases. As rrr decreases, we lose gravitational potential energy - in other words, U_GU
G
U, start subscript, G, end subscript becomes more negative. Because energy is conserved, the velocity must increase, resulting in an increase in kinetic energy.
Common mistakes and misconceptions
Students forget that there must be two separated objects considered as the system to have potential energy. A single object cannot have potential energy with itself, but only with respect to another object. For example, the Moon only has gravitational potential energy relative to the Earth (or another object).
Sometimes people forget that gravitational potential energy at large distances is negative. We typically make a choice for the location of our zero point of gravitational potential energy at a distance rrr of infinity. This makes all values of the gravitational potential energy negative.
Explanation: