In this question, we are asked to find the ratio of the linear velocities of a planet at two points in its elliptical orbit: when it is closest to the sun (at distance r₁) and when it is farthest from the sun (at distance r₂).
Kepler’s laws of planetary motion
According to Kepler’s second law (the law of areas), the line connecting the sun and the planet sweeps out equal areas in equal intervals of time. This implies that the planet moves faster when it is closer to the sun (at r₁) and slower when it is farther away (at r₂). Therefore, the velocity is inversely proportional to the distance from the sun at each point in the orbit.
Concept of Conservation of Angular Momentum
In the absence of external forces, the angular momentum of the planet about the sun is conserved. The angular momentum L at any point in the orbit is given by:
L = m * v * r
Where:
m is the mass of the planet
v is the linear velocity
r is the distance from the sun
Since angular momentum is conserved, we can equate the angular momentum at two different points in the orbit:
m * v₁ * r₁ = m * v₂ * r₂
The mass of the planet (m) cancels out, and we are left with:
v₁ * r₁ = v₂ * r₂
Solving for the velocity ratio v₁/v₂:
v₁/v₂ = r₂/r₁
Conclusion:
The ratio of the velocities is the inverse of the ratio of the distances from the sun. This matches option (D) in the question.
So, the correct answer is:
(D) (r₂/r₁)²
