Two satellites S₁ and S₂ are revolving in circular orbits around a planet with radius R₁ = 3200 km and R₂ = 800 km respectively. The ratio of speed of satellite S₁ to be speed of satellite S₂ in their respective orbits would be 1/x where x = ___________.

Solution

Physics sir · 3 steps

Orbital speed goes as one over root of the radius, so a radius four times bigger halves the speed.

1. Gravity gives centripetal force. For a satellite of mass m in a circular orbit of radius R around a planet of mass M, gravity supplies the centripetal force: GMm/R² = mv²/R.

   v = √(GM/R) ⇒ v₁/v₂ = √(R₂/R₁)

2. Put in numbers. Both satellites orbit the same planet, so GM cancels in the ratio. Then v₁/v₂ = √(800/3200) = √(1/4) = 1/2.
3. So. The ratio is 1/x = 1/2, which gives x = 2.

The common slip: 4. Dropping the square root and taking v ∝ 1/R gives a ratio of 1/4, so x = 4 instead of 2.

The idea to keep. For circular orbits around the same body, v ∝ 1/√R. The satellite in the farther orbit always moves slower, and the ratio needs a square root.

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