Chapter 5, Problem 11P

To determine

The orbital speeds at surfaces of Mercury, Venus, the Moon, and Mars.

Answer to Problem 11P

The orbital speeds at surfaces of Mercury, Venus, the Moon, and Mars are $2996\text{m/s}$, $7323\text{m/s}$, $1679\text{m/s}$, and $3542\text{m/s}$.

Explanation of Solution

Write the relation for the orbital speed at the surface of planet.

    $V_{\text{c}}=\sqrt{\frac{GM}{r}}$        (1)

Here, $V_{\text{c}}$ is orbital speed at the surface of the planet, $G$ is gravitational constant, $M$ is mass of the planet, $r$ is radius of the orbit.

In this case, the radius of the orbit is equal to the radius of the planet.

The mass of earth is $\left(M_{\oplus }=5.97\times {10}^{24}\text{kg}\right)$

The orbital speed at the surface of Mercury:

Substitute $0.055M_{\oplus }$ for $M$, $6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}$ for $G$, $2440\text{km}$ for $r$ in equation (1) to find $V_{\text{c}}$.

    $\begin{array}{c}{V}_{\text{c}}=\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.055M_{\oplus }\right)}{2440\text{km}}}\\ =\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.055\times 5.97\times {10}^{24}\text{kg}\right)}{\left(2440\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)}}\\ \approx 2996\text{m/s}\end{array}$

The orbital speed at the surface of Venus:

Substitute $0.815M_{\oplus }$ for $M$, $6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}$ for $G$, $6052\text{km}$ for $r$ in equation (1) to find $V_{\text{c}}$.

    $\begin{array}{c}{V}_{\text{c}}=\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.815M_{\oplus }\right)}{6052\text{km}}}\\ =\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.815\times 5.97\times {10}^{24}\text{kg}\right)}{\left(6052\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)}}\\ \approx 7323\text{m/s}\end{array}$

The orbital speed at the surface of the Moon:

Substitute $7.35\times {10}^{22}\text{kg}$ for $M$, $6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}$ for $G$, $1.74\times {10}^3\text{km}$ for $r$ in equation (1) to find $V_{\text{c}}$.

    $\begin{array}{c}{V}_{\text{c}}=\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(7.35\times {10}^{22}\text{kg}\right)}{1.74\times {10}^3\text{km}}}\\ =\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(7.35\times {10}^{22}\text{kg}\right)}{\left(1.74\times {10}^3\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)}}\\ \approx 1679\text{m/s}\end{array}$

The orbital speed at the surface of Mars:

Substitute $0.107M_{\oplus }$ for $M$, $6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}$ for $G$, $3396\text{km}$ for $r$ in equation (1) to find $V_{\text{c}}$.

    $\begin{array}{c}{V}_{\text{c}}=\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.107M_{\oplus }\right)}{3396\text{km}}}\\ =\sqrt{\frac{\left(6.67\times {10}^{-11}{\text{m}}^{\text{3}}{\text{/s}}^{\text{2}}\cdot \text{kg}\right)\left(0.107\times 5.97\times {10}^{24}\text{kg}\right)}{\left(3396\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)}}\\ \approx 3542\text{m/s}\end{array}$

Conclusion:

Therefore, the orbital speeds at surfaces of Mercury, Venus, the Moon, and Mars are $2996\text{m/s}$, $7323\text{m/s}$, $1679\text{m/s}$, and $3542\text{m/s}$.