Find the speeds of a planet of mass `m` in its perihelion and aphelion position. The semimajor axis of its orbit is a, eccentricity is `e` and the mass of the sun is `M`. Also find the total energy of the planet in terms of the given parameters.

Options

A. `E = - (GMm)/(2a)`
B. `E = - (GMm)/(4a)`
C. `E = - 3(GMm)/(2a)`
D. `E = - (GMm)/(3a)`

Explanation

To find the speeds of a planet at its perihelion and aphelion positions, as well as the total energy of the planet in terms of the given parameters, we can follow these steps:

### Step 1: Define the distances at perihelion and aphelion
- The distance from the Sun to the planet at perihelion (closest point) is given by:
\[ r_1 = a(1 - e) \]
- The distance from the Sun to the planet at aphelion (farthest point) is given by:
\[ r_2 = a(1 + e) \]

### Step 2: Use conservation of angular momentum
- The angular momentum \( L \) of the planet is conserved. At perihelion and aphelion, we have:
\[ L = m v_1 r_1 = m v_2 r_2 \]
This simplifies to:
\[ v_1 r_1 = v_2 r_2 \]
Rearranging gives:
\[ v_1 = v_2 \frac{r_2}{r_1} \]

### Step 3: Use conservation of energy
- The total mechanical energy \( E \) of the planet at perihelion and aphelion is given by:
\[ E = \text{Kinetic Energy} + \text{Potential Energy} \]
At perihelion:
\[ E = \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} \]
At aphelion:
\[ E = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \]
Setting these equal gives:
\[ \frac{1}{2} m v_1^2 - \frac{G M m}{r_1} = \frac{1}{2} m v_2^2 - \frac{G M m}{r_2} \]

### Step 4: Substitute \( r_1 \) and \( r_2 \) into the energy equation
- Substitute \( r_1 = a(1 - e) \) and \( r_2 = a(1 + e) \):
\[ \frac{1}{2} m v_1^2 - \frac{G M m}{a(1 - e)} = \frac{1}{2} m v_2^2 - \frac{G M m}{a(1 + e)} \]

### Step 5: Solve for \( v_1 \) and \( v_2 \)
- Rearranging the energy equation gives:
\[ \frac{1}{2} m v_1^2 - \frac{1}{2} m v_2^2 = \frac{G M m}{a(1 + e)} - \frac{G M m}{a(1 - e)} \]
- Factor out \( \frac{G M m}{a} \):
\[ \frac{1}{2} m (v_1^2 - v_2^2) = \frac{G M m}{a} \left( \frac{1}{1 + e} - \frac{1}{1 - e} \right) \]
- Solve for \( v_1 \) and \( v_2 \):
\[ v_1 = \sqrt{\frac{G M}{a} \cdot \frac{1 + e}{1 - e}}, \quad v_2 = \sqrt{\frac{G M}{a} \cdot \frac{1 - e}{1 + e}} \]

### Step 6: Calculate the total energy
- The total energy \( E \) can be expressed as:
\[ E = \frac{1}{2} m v^2 - \frac{G M m}{r} \]
- Using the average distance \( r \) in an elliptical orbit, we find:
\[ E = -\frac{G M m}{2a} \]

### Final Results
- The speeds at perihelion and aphelion are:
\[ v_1 = \sqrt{\frac{G M}{a} \cdot \frac{1 + e}{1 - e}}, \quad v_2 = \sqrt{\frac{G M}{a} \cdot \frac{1 - e}{1 + e}} \]
- The total energy of the planet is:
\[ E = -\frac{G M m}{2a} \]