Question

Write an expression for the total energy at aphelion in terms of the variables characterizing the motion there (i.e., vₐ and rₐ). Express your answer in terms of some or all of the variables vₐ, rₐ, and some or all of the fixed constants m, M, and g.

a) E_total = -G * (m * M) / (2 * rₐ) + (1/2) * m * vₐ^2 - G * (m * M) / rₐ
b) E_total = G * (m * M) / (2 * rₐ) - (1/2) * m * vₐ^2 + G * (m * M) / rₐ
c) E_total = -G * (m * M) / rₐ + (1/2) * m * vₐ^2 - G * (m * M) / (2 * rₐ)
d) E_total = G * (m * M) / rₐ - (1/2) * m * vₐ^2 + G * (m * M) / (2 * rₐ)

Answer 1

The total energy at aphelion is the sum of the potential and kinetic energies, mathematically expressed as E_total = -G * (m * M) / ra + (1/2) * m * va^2, which simplifies to E_total = -G * (m * M) / (2 * ra).

Step-by-step explanation:

To write an expression for the total energy at aphelion, we need to consider both potential and kinetic energies. The potential energy (U) at aphelion, due to the gravitational attraction between the two masses m (mass of the satellite) and M (mass of the central body) separated by a distance ra (aphelion distance), is given by -G * (m * M) / ra, where G is the gravitational constant. Kinetic energy (K) at aphelion, where the satellite has velocity va, is given by (1/2) * m * va2. Therefore, the total energy (Etotal) at aphelion, the sum of kinetic and potential energies, is Etotal = -G * (m * M) / ra + (1/2) * m * va2, which can be re-expressed using the fact that for any orbit, the total energy Etotal is one-half the negative of the potential energy at aphelion, leading to Etotal = -G * (m * M) / (2 * ra).