If the mass of Mars is 0.107 `M_(E )` and its radius is 0.53 `R_(E )`, estimate the gravitational field g at the surface of Mars.

Explanation

To estimate the gravitational field \( g \) at the surface of Mars, we can use the formula for gravitational field strength:

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

where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

Given:
- The mass of Mars \( M_{Mars} = 0.107 \cdot M_{E} \) (where \( M_{E} \) is the mass of Earth),
- The radius of Mars \( R_{Mars} = 0.53 \cdot R_{E} \) (where \( R_{E} \) is the radius of Earth).

### Step 1: Substitute the values into the formula

We can express the gravitational field \( g_{Mars} \) at the surface of Mars as:

\[ g_{Mars} = \frac{G \cdot (0.107 \cdot M_{E})}{(0.53 \cdot R_{E})^2} \]

### Step 2: Simplify the equation

Now, we simplify the denominator:

\[ (0.53 \cdot R_{E})^2 = 0.53^2 \cdot R_{E}^2 = 0.2809 \cdot R_{E}^2 \]

Thus, we can rewrite the equation for \( g_{Mars} \):

\[ g_{Mars} = \frac{G \cdot (0.107 \cdot M_{E})}{0.2809 \cdot R_{E}^2} \]

### Step 3: Relate it to Earth's gravitational field

We know that the gravitational field at the surface of Earth is given by:

\[ g_{E} = \frac{G \cdot M_{E}}{R_{E}^2} \]

We can express \( g_{Mars} \) in terms of \( g_{E} \):

\[ g_{Mars} = \frac{0.107 \cdot G \cdot M_{E}}{0.2809 \cdot R_{E}^2} = \frac{0.107}{0.2809} \cdot g_{E} \]

### Step 4: Calculate the value

Substituting \( g_{E} \approx 9.8 \, \text{m/s}^2 \):

\[ g_{Mars} = \frac{0.107}{0.2809} \cdot 9.8 \]

Calculating \( \frac{0.107}{0.2809} \):

\[ \frac{0.107}{0.2809} \approx 0.381 \]

Now, substituting this back:

\[ g_{Mars} \approx 0.381 \cdot 9.8 \approx 3.73 \, \text{m/s}^2 \]

### Final Answer

Thus, the gravitational field \( g \) at the surface of Mars is approximately:

\[ g_{Mars} \approx 3.73 \, \text{m/s}^2 \]

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