The escape velocity for the earth is 11.2 km / sec . The mass of another planet is 100 times that of the earth and its radius is 4 times that of the earth. The escape velocity for this planet will be

Options

A. 112.0 km / s
B. 5.6 km / s
C. 280.0 km / s
D. 56.0 km / s

Answer

To find the escape velocity for the planet, we can use the formula for escape velocity:

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

Where:
- \( v \) is the escape velocity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 1: Write the escape velocity for Earth
For Earth, the escape velocity \( v_E \) is given as:

\[ v_E = \sqrt{\frac{2GM_E}{R_E}} \]

Where \( M_E \) is the mass of the Earth and \( R_E \) is the radius of the Earth.

### Step 2: Write the escape velocity for the other planet
For the other planet, we know:
- Its mass \( M_P = 100 M_E \) (100 times the mass of Earth),
- Its radius \( R_P = 4 R_E \) (4 times the radius of Earth).

Now, substituting these values into the escape velocity formula for the planet:

\[ v_P = \sqrt{\frac{2G(100M_E)}{4R_E}} \]

### Step 3: Simplify the expression
We can simplify the expression for \( v_P \):

\[ v_P = \sqrt{\frac{200GM_E}{4R_E}} = \sqrt{\frac{50GM_E}{R_E}} \]

### Step 4: Relate it to Earth's escape velocity
Notice that we can express \( \frac{50GM_E}{R_E} \) in terms of \( v_E \):

\[ v_P = \sqrt{25 \cdot \frac{2GM_E}{R_E}} = \sqrt{25} \cdot \sqrt{\frac{2GM_E}{R_E}} = 5 \cdot v_E \]

### Step 5: Substitute the value of \( v_E \)
Now, substituting the value of \( v_E = 11.2 \, \text{km/s} \):

\[ v_P = 5 \cdot 11.2 \, \text{km/s} = 56 \, \text{km/s} \]

### Final Answer
Thus, the escape velocity for the planet is:

\[ \boxed{56 \, \text{km/s}} \]