Question

The ratio of KE required to be given to the satellite to escape earth's gravitational field to the KE required to be given, so that the satellite moves in a circular orbit just above earth's atmosphere is :

• A. 1
• B. \( \frac{1}{2} \)
• C. 2
• D. infinity

Hint:

Escape velocity is \(\sqrt{2}\) times orbital velocity $\Rightarrow$ KE ratio = 2.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We compare the kinetic energy needed to reach escape velocity from the surface with the kinetic energy needed to maintain a circular orbital velocity at the same radius.
: Key Formula or Approach:
1. Escape velocity \( v_e = \sqrt{2gR} \).
2. Orbital velocity \( v_o = \sqrt{gR} \).
3. Kinetic Energy \( K = \frac{1}{2}mv^2 \).
Step 2: Detailed Explanation:
- Kinetic energy required for escape (\( K_e \)):
\[ K_e = \frac{1}{2} m v_e^2 = \frac{1}{2} m (\sqrt{2gR})^2 = mgR \]
- Kinetic energy required for circular orbit (\( K_o \)):
\[ K_o = \frac{1}{2} m v_o^2 = \frac{1}{2} m (\sqrt{gR})^2 = \frac{1}{2} mgR \]
- Calculating the ratio \( K_e / K_o \):
\[ \text{Ratio} = \frac{mgR}{\frac{1}{2} mgR} = 2 \]
Step 3: Final Answer:
The ratio of the kinetic energies is 2.