Question:
If the radius of a planet is three times the radius of the earth. Both have same mass-densities. \( v_P \) and \( v_E \) are the escape velocities of the planet and the earth respectively, then
If the radius of a planet is three times the radius of the earth. Both have same mass-densities. \( v_P \) and \( v_E \) are the escape velocities of the planet and the earth respectively, then
Show Hint
For same density planets: \( v \propto R \). Bigger planet $\longrightarrow$ higher escape velocity.
Updated On: Apr 22, 2026
- \( v_P = 1.5v_E \)
- \( v_P = 3v_E \)
- \( v_E = 2v_P \)
- \( v_P = 2v_E \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Escape velocity:
\[
v = \sqrt{\frac{2GM}{R}}
\]
For same density:
\[
M \propto R^3
\]
Step 1: Relate mass and radius.
\[ \frac{M_P}{M_E} = \left(\frac{R_P}{R_E}\right)^3 = 3^3 = 27 \]
Step 2: Use escape velocity formula.
\[ v \propto \sqrt{\frac{M}{R}} \] \[ \frac{v_P}{v_E} = \sqrt{\frac{M_P/R_P}{M_E/R_E}} = \sqrt{\frac{27/3}{1}} = \sqrt{9} = 3 \]
Step 3: Final relation.
\[ v_P = 3v_E \]
Step 1: Relate mass and radius.
\[ \frac{M_P}{M_E} = \left(\frac{R_P}{R_E}\right)^3 = 3^3 = 27 \]
Step 2: Use escape velocity formula.
\[ v \propto \sqrt{\frac{M}{R}} \] \[ \frac{v_P}{v_E} = \sqrt{\frac{M_P/R_P}{M_E/R_E}} = \sqrt{\frac{27/3}{1}} = \sqrt{9} = 3 \]
Step 3: Final relation.
\[ v_P = 3v_E \]
Was this answer helpful?
0
0
Top MET Physics Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- A rope of mass $0.1\, kg$ is connected at the same height of two opposite walls. It is allowed to hang under its own weight. At [he contact point between the rope and the wall, the rope makes an angle $ \theta =10^{\circ} $ with respect to horizontal. The tension in the rope at its midpoint between the walls is
- At great distances from an electric dipole, the electric field strength due to the dipole varies with the distance as
View More Questions
Top MET Motion in a straight line Questions
- From Kepler's third law, the square of the period of revolution ($T$) is proportional to the cube of the semi-major axis ($R$). If a planet has $T_{p = 27T_{e}$, then:
- If the length of a simple pendulum is increased by 44%, the percentage increase in its time period is:
- The time period of a satellite of earth is 5 hours. If the separation between the earth and the satellite is increased to 4 times the previous value, the new time period will become:
- The one which decreases with dilution is
- The rms velocity of molecules of a gas of density $4~\text{kg m}^{-3}$ and pressure $1.2 \times 10^{5}~\text{N m}^{-2}$ is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions