Question

If \( r_1, v_1, L_1 \) and \( r_2, v_2, L_2 \) are radii, velocities, and angular momenta of a planet at perihelion and aphelion of its elliptical orbit around the Sun respectively, then

• A. \( r_1 v_1 = r_2 v_2, \, L_1 = L_2 \)
• B. \( r_1 v_1 = r_2 v_2, \, L_1 \neq L_2 \)
• C. \( r_1 v_1 \neq r_2 v_2, \, L_1 = L_2 \)
• D. \( r_1 v_1 \neq r_2 v_2, \, L_1 \neq L_2 \)

Hint:

In elliptical orbits, remember that \( r v \) is constant, but angular momentum \( L \) can vary depending on the position of the planet in its orbit.

Answer 1

The Correct Option is B

In elliptical orbits, according to the conservation of angular momentum, we have the relation: \[ L = m r v \] Where \( L \) is the angular momentum, \( m \) is the mass of the planet, \( r \) is the radius, and \( v \) is the velocity. Angular momentum is conserved in the orbit, and thus: \[ L_1 = L_2 \Rightarrow m r_1 v_1 = m r_2 v_2 \] Since the mass \( m \) is the same, we can cancel it out and get: \[ r_1 v_1 = r_2 v_2 \] This shows that the product of the radius and velocity is constant throughout the orbit, even though the velocity at perihelion and aphelion might differ. However, the total angular momentum at different points may vary due to the variation in the distance of the planet from the Sun, meaning \( L_1 \neq L_2 \). Therefore, the correct option is: \[ r_1 v_1 = r_2 v_2, \, L_1 \neq L_2 \]