A planet revolving in elliptical orbit has :
A. a constant velocity of revolution.
B. has the least velocity when it is nearest to the sun.
C. its areal velocity is directly proportional to its velocity.
D. areal velocity is inversely proportional to its velocity.
E. to follow a trajectory such that the areal velocity is constant.
Choose the correct answer from the options given below :

Solution

Physics sir · 3 steps

The sun exerts no torque on the planet, so angular momentum and areal velocity both stay fixed.

1. No torque. The sun's pull always points along the line joining the planet to the sun. A force along r⃗ gives zero torque about the sun, so the planet's angular momentum L stays constant.

   dA/dt = 1/2 r v_⊥ = L/2m = constant

2. Speed must change. Because r v_⊥ is fixed, a smaller r means a larger speed. The planet moves fastest when nearest the sun (perihelion) and slowest when farthest (aphelion).

   Statement	True or false	Reason
   A. Constant velocity	False	Speed changes with r, and the direction changes all the time
   B. Least speed nearest the sun	False	Nearest the sun r is smallest, so speed is greatest
   C. Areal velocity ∝ speed	False	Areal velocity does not change at all while speed does
   D. Areal velocity ∝ 1/speed	False	Same reason: areal velocity is constant, independent of speed
   E. Areal velocity constant	True	Kepler's second law, from conservation of L

3. So. Only statement E is correct, so the answer is (b) E only.

The idea to keep. Gravity acts along the line to the sun, so L is conserved and dA/dt = L/2m is constant. Speed is not constant: fastest at perihelion, slowest at aphelion.

Answer from the NTA final answer key. Solution by PrepWiser.