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Chapter 5: Q5-64P (page 109)

(II) Planet A and planet B are in circular orbits around a distant star. Planet A is 7.0 times farther from the star than is planet B. What is the ratio of their speeds \({v_A}/{v_B}\)?

Short Answer

Expert verified

The ratio of their speeds is \(\sqrt 7 \).

Step by step solution

01

Concept

When a body moves on path from a fixed distance, then the body is said to be orbiting around the center.

02

Given data

The relation between the distance from the star is \({r_B} = 7{r_A}\).

03

Calculation

The expression for the orbital velocity is given by,

\(v = \sqrt {\frac{{GM}}{r}} \)

Here, G is the gravitational constant, M is the mass, and r is the distance from center.

For two different positions above equation can be written as,

\(\frac{{{v_A}}}{{{v_B}}} = \sqrt {\frac{{{r_B}}}{{{r_A}}}} \)

Substitute the given condition in above equation,

\(\begin{aligned}\frac{{{v_A}}}{{{v_B}}} &= \sqrt {\frac{{7{r_A}}}{{{r_A}}}} \\\frac{{{v_A}}}{{{v_B}}} &= \sqrt 7 \end{aligned}\)

Thus, the ratio of their speeds is \(\sqrt 7 \).

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