To find the gravitational acceleration $g$ on the spacecraft orbiting Mars, we use the gravitational formula:

$g = \frac{GM}{r^2}$

where:

• $G$ is the gravitational constant, $6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2$,
• $M$ is the mass of Mars, $6.39 \times 10^{23} \, \text{kg}$,
• $r$ is the distance from the center of Mars to the spacecraft. Since the spacecraft is 3.92 × $10^5 \, \text{m}$ above Mars's surface and Mars's radius is $3.40 \times 10^6 \, \text{m}$, the total distance $r$ is:

$r = 3.40 \times 10^6 + 3.92 \times 10^5 = 3.792 \times 10^6 \, \text{m}$

Calculation

Now, substitute the values into the formula:

$g = \frac{(6.674 \times 10^{-11})(6.39 \times 10^{23})}{(3.792 \times 10^6)^2}$

Let's calculate this step-by-step:

1. Calculate the numerator: $6.674 \times 10^{-11} \times 6.39 \times 10^{23}$.
2. Calculate the denominator: $(3.792 \times 10^6)^2$.
3. Divide the result from step 1 by the result from step 2 to get $g$.

I'll compute this for you.The gravitational acceleration $g$ on the spacecraft orbiting Mars is approximately $2.97 \, \text{m/s}^2$.

Would you like a breakdown of the calculations or have any questions?

Here are some related questions to consider:

1. How would the gravitational force change if the spacecraft was further from Mars?
2. What would the gravitational acceleration be if the spacecraft were at Mars's surface?
3. How does Mars's gravitational acceleration compare to Earth's at the same altitude?
4. How would the mass of the spacecraft affect its orbit around Mars?
5. What orbital velocity is required for the spacecraft to remain in orbit at this altitude?

Tip: Remember that gravitational force decreases with the square of the distance, so altitude changes significantly affect $g$.