Question:
Two planets, A and B orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is:
Two planets, A and B orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is:
Show Hint
For planets revolving around the same star:
\[
T^2\propto r^3
\]
and
\[
v\propto \frac{1}{\sqrt{r}}.
\]
First find the orbital radius ratio using Kepler’s law, then calculate the velocity ratio.
Updated On: Jun 24, 2026
- \(4:1\)
- \(1:4\)
- \(2:1\)
- \(1:2\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Use Kepler’s third law.
For planets revolving around the same star, \[ T^2\propto r^3 \] where \[ T=\text{time period} \] and \[ r=\text{orbital radius} \]
Step 2: Write the ratio of time periods.
Given, \[ T_A=8T_B \] Thus, \[ \frac{T_A}{T_B}=8 \] Using Kepler’s law, \[ \left(\frac{T_A}{T_B}\right)^2 = \left(\frac{r_A}{r_B}\right)^3 \] \[ 8^2=\left(\frac{r_A}{r_B}\right)^3 \] \[ 64=\left(\frac{r_A}{r_B}\right)^3 \]
Step 3: Find the ratio of orbital radii.
Taking cube root, \[ \frac{r_A}{r_B}=4 \] Thus, \[ r_A:r_B=4:1 \]
Step 4: Use the formula for orbital velocity.
Orbital velocity is \[ v=\sqrt{\frac{GM}{r}} \] Therefore, \[ v\propto \frac{1}{\sqrt{r}} \]
Step 5: Find the ratio of orbital velocities.
\[ \frac{v_A}{v_B} = \sqrt{\frac{r_B}{r_A}} \] \[ = \sqrt{\frac{1}{4}} \] \[ =\frac{1}{2} \] Thus, \[ v_A:v_B=1:2 \]
Step 6: Final conclusion.
Hence, the required ratio is \[ \boxed{1:2} \]
For planets revolving around the same star, \[ T^2\propto r^3 \] where \[ T=\text{time period} \] and \[ r=\text{orbital radius} \]
Step 2: Write the ratio of time periods.
Given, \[ T_A=8T_B \] Thus, \[ \frac{T_A}{T_B}=8 \] Using Kepler’s law, \[ \left(\frac{T_A}{T_B}\right)^2 = \left(\frac{r_A}{r_B}\right)^3 \] \[ 8^2=\left(\frac{r_A}{r_B}\right)^3 \] \[ 64=\left(\frac{r_A}{r_B}\right)^3 \]
Step 3: Find the ratio of orbital radii.
Taking cube root, \[ \frac{r_A}{r_B}=4 \] Thus, \[ r_A:r_B=4:1 \]
Step 4: Use the formula for orbital velocity.
Orbital velocity is \[ v=\sqrt{\frac{GM}{r}} \] Therefore, \[ v\propto \frac{1}{\sqrt{r}} \]
Step 5: Find the ratio of orbital velocities.
\[ \frac{v_A}{v_B} = \sqrt{\frac{r_B}{r_A}} \] \[ = \sqrt{\frac{1}{4}} \] \[ =\frac{1}{2} \] Thus, \[ v_A:v_B=1:2 \]
Step 6: Final conclusion.
Hence, the required ratio is \[ \boxed{1:2} \]
Was this answer helpful?
0
0
Top AP EAPCET Physics Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The radii of the two columns in a U - tube are r1 and r2. When a liquid of density p (angle of contact is $ 0{}^\circ $ ) is filled in it, the level difference of the liquid in the two arms is h. The surface tension of the liquid is: (g = acceleration due to gravity)
- A rigid metallic sphere is spinning around its own axis in the absence of external torque. If the temperature is raised, its volume increases by $9 \%$. The change in its angular speed is
View More Questions
Top AP EAPCET Keplers Laws Questions
- Two planets, A and B, orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is
- The figure shows the revolution of the planet (P) around the sun (S) in an elliptical orbit. Which of the following statements is true regarding time taken?
- A planet revolves around the sun in an elliptical orbit. The areal velocity is 4 10^16 m^2s^-1. Maximum distance is 4 10^12 m. Find minimum speed.
- A planet revolves around the Sun in an elliptical orbit. The areal velocity of the planet is \(4 \times 10^{16}\,\text{m}^2\text{s}^{-1}\). If the maximum distance between the planet and the Sun is \(4 \times 10^{12}\,\text{m}\), then the minimum speed of the planet is:
- Assume a planet with orbiting radius R and period of revolution T around the sun experiences a gravitational force which follows inverse cube law instead of inverse square law. In this case, the period of revolution is proportional to:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions