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Chapter 6: Problem 791

A planet moving along an elliptical orbit is closest to the sun at a distance \(\mathrm{r}_{1}\) and farthest away at a distance of \(\mathrm{r}_{2}\). If \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) are the liner velocities at these points respectively, then the ratio \(\left(\mathrm{v}_{1} / \mathrm{v}_{2}\right)\) is \(\ldots \ldots \ldots \ldots\) (A) \(\left(\mathrm{r}_{1} / \mathrm{r}_{2}\right)\) (B) \(\left(\mathrm{r}_{1} / \mathrm{r}_{2}\right)^{2}\) (C) \(\left(\mathrm{r}_{2} / \mathrm{r}_{1}\right)\) (D) \(\left(\mathrm{r}_{2} / \mathrm{r}_{1}\right)^{2}\)

Short Answer

Expert verified
The short answer is: \(\frac{v_1}{v_2} = \frac{r_2}{r_1}\) (C)

Step by step solution

01

Write down Kepler's second law formula

According to Kepler's second law, the area swept out per unit of time, A/t, is constant in the elliptical orbit. For each point (r1 and r2) in the orbit, we can express the area per time as: \[ \frac{1}{2}rv = \text{const}\] Where r is the distance from the sun, v is the linear velocity, and the constant is the same for both points in the orbit.
02

Apply Kepler's second law for r1 and r2

Using the formula given in step 1, we can write the constant for both closest (r1, v1) and farthest (r2, v2) distances from the Sun: \[ \frac{1}{2}r_1v_1 = \frac{1}{2}r_2v_2\]
03

Solve for the ratio v1/v2

Now, we can solve this equation for the ratio v1/v2, by dividing both sides of the equation by r1*v1, and then by r2*v2, which leads to: \[\frac{v_1}{v_2} = \frac{r_2}{r_1}\]
04

Choose the correct answer

We can now see that the correct answer to the problem is: \(\frac{v_1}{v_2} = \frac{r_2}{r_1}\) which corresponds to choice (C).

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Most popular questions from this chapter

A body of mass \(\mathrm{m} \mathrm{kg}\) starts falling from a point $2 \mathrm{R}\( above the earth's surface. Its \)\mathrm{K} . \mathrm{E}$. when it has fallen to a point ' \(\mathrm{R}\) ' above the Earth's surface $=\ldots \ldots \ldots \ldots J$ [R - Radius of Earth, M-mass of Earth G-Gravitational constant \(]\) (A) \((1 / 2)[(\mathrm{GMm}) / \mathrm{R}]\) (B) \((1 / 6)[(\mathrm{GMm}) / \mathrm{R}]\) (C) \((2 / 3)[(\mathrm{GMm}) / \mathrm{R}]\) (D) \((1 / 3)[(\mathrm{GMm}) / \mathrm{R}]\)

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