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Chapter 8: Problem 75

Two satellites are moving around the earth in circular orbits at height \(R\) and \(3 R\) respectively, \(R\) being the radius of the earth, the ratio of their kinetic energies is (a) 2 (b) 4 (c) 8 (d) 16

Short Answer

Expert verified
The ratio of their kinetic energies is 2, hence option (a) is the correct answer.

Step by step solution

01

Calculate orbital velocity of first satellite

Let's denote the first satellite as 1 and the second satellite as 2. The orbital velocity of the first satellite (v1) is given by \( v1 = \sqrt{\frac{G M}{R + R}} \).
02

Calculate orbital velocity of second satellite

Similarly, the orbital velocity of the second satellite (v2) is given by \( v2 = \sqrt{\frac{G M}{R + 3R}} \).
03

Calculate kinetic energy of first satellite

The kinetic energy of the first satellite (KE1) is given by \( KE1 = \frac{1}{2} m v1^2 \). Substitute the expression for v1 from step 1.
04

Calculate kinetic energy of second satellite

In the same way, the kinetic energy of the second satellite (KE2) is given by \( KE2 = \frac{1}{2} m v2^2 \). Substitute the expression for v2 from step 2.
05

Find the ratio of their kinetic energies

The ratio of their kinetic energies is given by \( \frac{KE1}{KE2} = \frac{v1^2}{v2^2} \). Substitute the expressions for v1 and v2 from steps 1 and 2 respectively. Simplify the equation which will result in a numerical ratio.

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