Radius of earth is 6400 km and that of mars is 3200 km. Mass of mars is 0.1 that of earth's mass. Then the acceleration due to gravity on mars is nearly

Options

A. `1 m//s^(2)`
B. `2.5 m//s^(2)`
C. `4 m//s^(2)`
D. `5 m//s^(2)`

Explanation

To find the acceleration due to gravity on Mars, we can use the formula for gravitational acceleration:

\[ g = \frac{G \cdot M}{R^2} \]

where:
- \( g \) is the acceleration due to gravity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet, and
- \( R \) is the radius of the planet.

### Step 1: Write down the known values
- Radius of Earth, \( R_e = 6400 \) km
- Radius of Mars, \( R_m = 3200 \) km
- Mass of Mars, \( M_m = 0.1 \cdot M_e \) (where \( M_e \) is the mass of Earth)

### Step 2: Write the formula for gravitational acceleration for Earth and Mars
The acceleration due to gravity on Earth is given by:

\[ g_e = \frac{G \cdot M_e}{R_e^2} \]

The acceleration due to gravity on Mars is given by:

\[ g_m = \frac{G \cdot M_m}{R_m^2} \]

### Step 3: Substitute the mass of Mars into the equation
Substituting \( M_m = 0.1 \cdot M_e \) into the equation for \( g_m \):

\[ g_m = \frac{G \cdot (0.1 \cdot M_e)}{R_m^2} \]

### Step 4: Divide the equations for \( g_m \) and \( g_e \)
To find the ratio of the gravitational accelerations on Mars and Earth, we can divide the equations:

\[ \frac{g_m}{g_e} = \frac{G \cdot (0.1 \cdot M_e) / R_m^2}{G \cdot M_e / R_e^2} \]

### Step 5: Simplify the equation
The \( G \) and \( M_e \) cancel out:

\[ \frac{g_m}{g_e} = \frac{0.1}{\left(\frac{R_m}{R_e}\right)^2} \]

### Step 6: Substitute the values of \( R_m \) and \( R_e \)
Substituting \( R_m = 3200 \) km and \( R_e = 6400 \) km:

\[ \frac{R_m}{R_e} = \frac{3200}{6400} = \frac{1}{2} \]

Thus,

\[ \left(\frac{R_m}{R_e}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 7: Substitute back into the equation
Now substituting back into the equation:

\[ \frac{g_m}{g_e} = \frac{0.1}{\frac{1}{4}} = 0.1 \times 4 = 0.4 \]

### Step 8: Calculate \( g_m \)
Assuming \( g_e \) (acceleration due to gravity on Earth) is approximately \( 10 \, \text{m/s}^2 \):

\[ g_m = 0.4 \cdot g_e = 0.4 \cdot 10 = 4 \, \text{m/s}^2 \]

### Conclusion
Thus, the acceleration due to gravity on Mars is approximately \( 4 \, \text{m/s}^2 \).