Question

Kepler's law of planetary motion says that the square of the period of a planet \(\left(T^{2}\right)\) is proportional to the cube of the distance of the planet from the Sun \(\left(r^{3}\right) .\) Mars is about twice as far from the Sun as Venus. How does the period of Mars compare with the period of Venus?

Short answer

Answer: The period of Mars is about 2.83 times the period of Venus.

Step-by-step solution

Write down Kepler's law

Kepler's law states that \(T^2 \propto r^3\), where \(T\) is the period of a planet, and \(r\) is the distance of the planet from the Sun.

Set up the proportion

Let \(T_{m}\) and \(r_{m}\) be the period and distance of Mars from the Sun, while \(T_{v}\) and \(r_{v}\) be the period and distance of Venus from the Sun. According to Kepler's law, we can set up the following proportions: $$\frac{T_{m}^2}{T_{v}^2} = \frac{r_{m}^3}{r_{v}^3}$$

Use given information

We are given that Mars is about twice as far from the Sun as Venus, which means \(r_{m} = 2r_{v}\). Substitute this into our proportion: $$\frac{T_{m}^2}{T_{v}^2} = \frac{(2r_{v})^3}{r_{v}^3}$$

Solve for the ratio of the periods

Simplify the proportion equation and solve for the ratio of the periods of Mars and Venus: $$\frac{T_{m}^2}{T_{v}^2} = \frac{8r_{v}^3}{r_{v}^3}$$ $$\frac{T_{m}^{2}}{T_{v}^{2}} = 8$$

Find the period of Mars in terms of Venus' period

To find how the period of Mars compares to the period of Venus, take the square root of both sides in the equation: $$\frac{T_{m}}{T_{v}} = \sqrt{8}$$ $$\frac{T_{m}}{T_{v}} \approx 2.83$$ So, the period of Mars is about 2.83 times the period of Venus.