A hypothetical planet of mass m is moving along an elliptical path around sun of mass `M_s` under the influence of its gravitational pull. If the major axis is 2R, find the speed of the planet when it is at a distance of R from the sun.

Options

A. `sqrt((GM_s)/R)`
B. `sqrt(2GM_s)/R`
C. `2sqrt((GM_s)/R`
D. `m/Msqrt(GM_s)/R`

Answer

To solve the problem of finding the speed of a hypothetical planet of mass \( m \) moving along an elliptical path around the sun of mass \( M_s \) when it is at a distance \( R \) from the sun, we can follow these steps:

### Step 1: Understand the elliptical orbit
The major axis of the elliptical orbit is given as \( 2R \). This means that the semi-major axis \( a \) of the ellipse is \( R \).

### Step 2: Use Kepler's Third Law
Kepler's Third Law states that the square of the period \( T \) of an orbiting body is proportional to the cube of the semi-major axis \( a \):
\[ T^2 = \frac{4\pi^2}{G M_s} a^3 \]
Substituting \( a = R \):
\[ T^2 = \frac{4\pi^2}{G M_s} R^3 \]
Taking the square root to find \( T \):
\[ T = 2\pi \sqrt{\frac{R^3}{G M_s}} \]

### Step 3: Determine the speed of the planet
The speed \( v \) of the planet at a distance \( R \) can be calculated using the formula for the orbital speed:
\[ v = \frac{2\pi r}{T} \]
where \( r \) is the distance from the sun, which is \( R \) in this case.

Substituting \( T \) from Step 2 into the speed formula:
\[ v = \frac{2\pi R}{2\pi \sqrt{\frac{R^3}{G M_s}}} \]
This simplifies to:
\[ v = \frac{R}{\sqrt{\frac{R^3}{G M_s}}} \]
\[ v = \frac{R \sqrt{G M_s}}{R^{3/2}} = \sqrt{\frac{G M_s}{R}} \]

### Final Result
Thus, the speed of the planet when it is at a distance \( R \) from the sun is:
\[ v = \sqrt{\frac{G M_s}{R}} \]