Question

If the ratio of the distances of a planet from the sun at perihelion and aphelion is 2: 3, then the ratio of their respective velocities at the perihelion and aphelion is

• A. 2:3
• B. 2:5
• C. 3:5
• D. 3:2
• E. 1:1

Hint:

Logic Tip: Velocity and orbital radius are inversely proportional at the apsides (perihelion and aphelion). The planet moves fastest when it is closest to the sun, and slowest when it is furthest. Simply flip the distance ratio to get the velocity ratio!

Answer 1

The Correct Option is D

Concept:
According to Kepler's Second Law (the law of areas), the angular momentum of a planet around the Sun is conserved because the gravitational force is a central force (producing zero torque). The angular momentum $L$ is given by $L = mvr\sin\theta$. At perihelion and aphelion, the velocity vector is strictly perpendicular to the position vector ($\theta = 90^\circ$), simplifying the conservation equation to: $$m v_p r_p = m v_a r_a$$
Step 1: Identify the given ratio.
Let $r_p$ be the distance at perihelion and $r_a$ be the distance at aphelion. We are given the ratio: $$\frac{r_p}{r_a} = \frac{2}{3}$$
Step 2: Apply Conservation of Angular Momentum.
$$m v_p r_p = m v_a r_a$$ Cancel the mass $m$ of the planet from both sides: $$v_p r_p = v_a r_a$$
Step 3: Solve for the velocity ratio.
Rearrange the equation to find the ratio of velocities $\frac{v_p}{v_a}$: $$\frac{v_p}{v_a} = \frac{r_a}{r_p}$$ Since $\frac{r_p}{r_a} = \frac{2}{3}$, its reciprocal is: $$\frac{v_p}{v_a} = \frac{3}{2}$$ Thus, the ratio is 3:2.