Question

a) By what percentage does the gravitational potential energy of the Earth change between perihelion and aphelion? (Assume that the Earth's potential energy would be zero if it moved to a very large distance away from the Sun.) b) By what percentage does the kinetic energy of the Earth change between perihelion and aphelion?

Short answer

Question: Calculate the percentage change in gravitational potential energy (a) and kinetic energy (b) of the Earth between perihelion and aphelion.

Step-by-step solution

Understand the terms

In this exercise, we have two terms perihelion and aphelion. Perihelion is the point in the orbit of a planet where it is closest to the Sun. Aphelion is the point in the orbit of a planet where it is furthest from the Sun. The Earth's distance from the Sun at perihelion is about 147.1 million kilometers, and at aphelion, it is about 152.1 million kilometers.

Calculate gravitational potential energy

The formula for gravitational potential energy (U) is: U = -G * (m1 * m2) / r where G is the gravitational constant, m1 and m2 are the masses of the two objects(Mass of Earth and Sun), and r is the distance between them. At perihelion (r1 = 147.1 million km or 147.1 * 10^9 m): U1 = -G * (Mass of Earth * Mass of Sun) / r1 At aphelion (r2 = 152.1 million km or 152.1 * 10^9 m): U2 = -G * (Mass of Earth * Mass of Sun) / r2

Calculate the percentage change in gravitational potential energy

To calculate the percentage change in gravitational potential energy, use the following formula: Percentage change = ((U2 - U1) / U1) * 100 Plug the values of U1 and U2 calculated above and find the percentage change in gravitational potential energy between perihelion and aphelion.

Calculate kinetic energy

The formula for kinetic energy (K) is: K = 0.5 * m * v^2 where m is the mass of the object (Earth) and v is its orbital velocity (which varies at perihelion and aphelion). At perihelion, Earth's orbital velocity (v1) is the highest. Calculate its kinetic energy K1: K1 = 0.5 * Mass of Earth * v1^2 At aphelion, Earth's orbital velocity (v2) is the lowest. Calculate its kinetic energy K2: K2 = 0.5 * Mass of Earth * v2^2

Calculate the percentage change in the kinetic energy

To calculate the percentage change in kinetic energy, use the same formula as in step 3: Percentage change = ((K2 - K1) / K1) * 100 Plug the values of K1 and K2 calculated above and find the percentage change in kinetic energy between perihelion and aphelion. Answer: Now you have calculated the percentage change in gravitational potential energy (part a) and kinetic energy (part b) of the Earth between perihelion and aphelion.