The ratio of kinetic energy of a planet at perigee and apogee during its motion around the sun in eliptical orbit of ecentricity e is

Options

A. `1:e`
B. `(1+e)/(1-e)`
C. `((1+e)/(1-e))^(2)`
D. `((1-e)/(1+e))^(2)`

Answer

To find the ratio of the kinetic energy of a planet at perigee and apogee during its motion around the sun in an elliptical orbit of eccentricity \( e \), we can follow these steps:

### Step 1: Define the distances at perigee and apogee
The distances from the sun to the planet at perigee (\( r_p \)) and apogee (\( r_a \)) can be expressed in terms of the semi-major axis \( a \) and the eccentricity \( e \):
- At perigee:
\[ r_p = a(1 - e) \]
- At apogee:
\[ r_a = a(1 + e) \]

### Step 2: Use the conservation of angular momentum
The angular momentum \( L \) of the planet is conserved. Therefore, we can write:
\[ L = m \cdot v_p \cdot r_p = m \cdot v_a \cdot r_a \]
where \( v_p \) is the velocity at perigee and \( v_a \) is the velocity at apogee.

### Step 3: Express the ratio of velocities
From the conservation of angular momentum, we can derive the ratio of the velocities:
\[ \frac{v_p}{v_a} = \frac{r_a}{r_p} \]

### Step 4: Substitute the expressions for \( r_p \) and \( r_a \)
Substituting the expressions for \( r_p \) and \( r_a \):
\[ \frac{v_p}{v_a} = \frac{a(1 + e)}{a(1 - e)} = \frac{1 + e}{1 - e} \]

### Step 5: Find the ratio of kinetic energies
The kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2}mv^2 \]
Thus, the ratio of the kinetic energies at perigee and apogee is:
\[ \frac{KE_p}{KE_a} = \frac{\frac{1}{2}mv_p^2}{\frac{1}{2}mv_a^2} = \frac{v_p^2}{v_a^2} \]
Substituting the velocity ratio:
\[ \frac{KE_p}{KE_a} = \left(\frac{v_p}{v_a}\right)^2 = \left(\frac{1 + e}{1 - e}\right)^2 \]

### Final Result
Thus, the ratio of the kinetic energy of the planet at perigee to that at apogee is:
\[ \frac{KE_p}{KE_a} = \left(\frac{1 + e}{1 - e}\right)^2 \]