Determine the velocity required for a moving object $5.00 \times 10^{3} \mathrm{~m}$ above the surface of Mars to escape from Mars's gravity. The mass of Mars is $6.42 \times 10^{23} \mathrm{~kg}$, and its radius is $3.40 \times 10^{3} \mathrm{~m}$.

Step 1 ：We are given that the object is \(5.00 \times 10^{3} \mathrm{~m}\) above the surface of Mars. The mass of Mars is \(6.42 \times 10^{23} \mathrm{~kg}\), and its radius is \(3.40 \times 10^{3} \mathrm{~m}\).

Step 2 ：The escape velocity from a planet can be calculated using the formula: \(v = \sqrt{\frac{2GM}{r}}\) where \(v\) is the escape velocity, \(G\) is the gravitational constant \(6.674 \times 10^{-11} \mathrm{~m}^3 \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\), \(M\) is the mass of the planet, and \(r\) is the distance from the center of the planet to the object.

Step 3 ：In this case, the object is \(5.00 \times 10^{3} \mathrm{~m}\) above the surface of Mars, so the distance from the center of Mars to the object is the radius of Mars plus the height of the object above the surface. Therefore, \(r = 3400.0 + 5000.0 = 8400.0\) m.

Step 4 ：Substituting the given values into the formula, we get \(v = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 6.42 \times 10^{23}}{8400.0}}\)

Step 5 ：Solving the above expression, we find that the escape velocity is approximately \(101003.39 \mathrm{~m/s}\).

Step 6 ：Final Answer: The velocity required for a moving object \(5.00 \times 10^{3} \mathrm{~m}\) above the surface of Mars to escape from Mars's gravity is approximately \(\boxed{101003.39 \mathrm{~m/s}}\).