# How does the conservation of mechanical energy explain why a planet in an elliptical...

## Question:

How does the conservation of mechanical energy explain why a planet in an elliptical or!@#\$%^&anges speed?

## Energy Conservation

Energy conservation is the principle that the energy that a system has remains the same over time, provided it does not transfer energy with another system. Energy may change from one form to another within a system but the total amount of energy in the system will not change.

An common example of energy conservation is a roller coaster. At the top of a drop the roller coaster is not moving, and thus has no kinetic energy but has a lot of gravitational potential energy in the form of its height. As the roller coaster makes its descent down the track that gravitational potential energy changes to kinetic energy, i.e. it changes its height to increase its speed. This is often used to solve many simple physics problems.

Kepler's first law of planetary motion states that planets orbiting the Sun do so in elliptical orbits. This was later shown by Newton to be a good approximation of objects orbiting any large body. As the planet traces out its elliptic path around the object its orbiting its distance from the object changes, and its speed changes accordingly. This is an example of conservation of mechanical energy. Let's consider the planet at two different points around its orbit, point 1 and point 2. Let's say the planet has mass m and the star has mass M.

At point 1 the object is a distance {eq}R_1 {/eq} from the star and has a velocity {eq}v_1 {/eq}. Due to it having a velocity we can say that it has a kinetic energy {eq}E_k = \frac{mv_1^2}{2} {/eq}, and the gravitational force between it and the star provides a gravitational potential energy {eq}E_g = -\frac{GMm}{R_1} {/eq} (G is the gravitational constant {eq}\approx 6.67 \times 10^{-11} m^3/kg/s^2 {/eq}). The planets total energy at point 1 is then given by:

{eq}E_1 = E_k+E_g =\frac{mv_1^2}{2}-\frac{GMm}{R_1} {/eq}

At point 2 the star has speed {eq}v_2 {/eq} (which may or may not be different from {eq}v_1 {/eq}) and is a distance {eq}R_2 {/eq} (which is different from {eq}R_1 {/eq}) away from the star. Therefore it also has a kinetic and potential energy, and thus total energy:

{eq}E_2 = E_k+E_g = \frac{mv_2^2}{2} - \frac{GMm}{R_2} {/eq}

Since conservation of energy must be obeyed by our planet/star system since it is not transferring energy to some other system, we know that these two total energies must be equal. Since the distance from the star has changed between our two cases something else must change to compensate for this, since the total energy can't change. Let's go over our options for what could change. G can't change since it is just a constant, neither can the masses of the planet or star, so we are just left with the velocity, which must change! Therefore, if the planet changes its distance from the star it is orbiting it must also change its velocity due to the conservation of mechanical energy. 