The minimum and maximum distances of a planet revolving around the sun are x₁ and x₂. If the minimum speed of the planet on its trajectory is v₀ then its maximum speed will be :

Solution

Physics sir · 3 steps

Gravity exerts no torque about the sun, so angular momentum at the nearest and farthest points must match.

1. Why L is conserved. The sun's pull on the planet always points along the line joining them, so it exerts no torque about the sun. The planet's angular momentum about the sun stays constant all along the orbit.
2. Pick the two points. At the nearest point (x₁) and the farthest point (x₂), the velocity is perpendicular to the radius, so L = mvr exactly there. The planet moves fastest when nearest and slowest when farthest, so the speed at x₂ is v₀.

   m vₘₐₓ x₁ = m v₀ x₂

3. So. vₘₐₓ = (v₀ x₂)/x₁. Since x₂ > x₁, this is larger than v₀, as it should be. The answer is (d).

The idea to keep. For any planet or satellite, use v₁ r₁ = v₂ r₂ only at the nearest and farthest points, where velocity is perpendicular to the radius. Speed goes as 1/r there, not 1/r².

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