Question

A planet has an escape speed of 10km/s,The radius of the planet is 10,000km. The acceleration due to gravity of the planet at its surface is:

• A. 10m/s²
• B. 9.8m/s²
• C. 20m/s²
• D. 2.5m/s²
• E. 5m/s²

Answer 1

Answer: (E)

Given:

• Escape speed, \( v_e = 10 \, \text{km/s} = 10,000 \, \text{m/s} \)
• Radius of planet, \( R = 10,000 \, \text{km} = 10^7 \, \text{m} \)

Step 1: Relate Escape Speed to Gravity

Escape speed is given by:

\[ v_e = \sqrt{\frac{2GM}{R}} \]

where \( G \) is the gravitational constant and \( M \) is the planet's mass.

Step 2: Express Surface Gravity

Acceleration due to gravity at the surface (\( g \)) is:

\[ g = \frac{GM}{R^2} \]

Step 3: Solve for \( g \) Using Escape Speed

Square the escape speed equation:

\[ v_e^2 = \frac{2GM}{R} \implies GM = \frac{v_e^2 R}{2} \]

Substitute \( GM \) into the gravity formula:

\[ g = \frac{v_e^2 R}{2 R^2} = \frac{v_e^2}{2R} \]

Plug in the given values:

\[ g = \frac{(10,000)^2}{2 \times 10^7} = \frac{10^8}{2 \times 10^7} = 5 \, \text{m/s}^2 \]

Conclusion:

The acceleration due to gravity at the planet's surface is 5 m/s\(^2\).

Answer: \(\boxed{E}\)

Answer 2

1. Recall the formula for escape speed:

The escape speed (v_e) of a planet is given by:

\[v_e = \sqrt{\frac{2GM}{R}}\]

where:

• G is the universal gravitational constant
• M is the mass of the planet
• R is the radius of the planet

2. Relate escape speed to acceleration due to gravity:

The acceleration due to gravity (g) at the surface of a planet is given by:

\[g = \frac{GM}{R^2}\]

3. Combine the equations and solve for g:

From the escape speed equation, we can write:

\[v_e^2 = \frac{2GM}{R}\]

Now, multiply both sides by R:

\[v_e^2 R = 2GM\]

Divide both sides by 2R²:

\[\frac{v_e^2}{2R} = \frac{GM}{R^2} = g\]

4. Plug in the given values:

We are given \(v_e = 10 \, km/s = 10^4 \, m/s\) and \(R = 10,000 \, km = 10^7 \, m\). Substituting these values into the equation for g:

\[g = \frac{(10^4 \, m/s)^2}{2 \times 10^7 \, m} = \frac{10^8 \, m^2/s^2}{2 \times 10^7 \, m} = 5 \, m/s^2\]