Escape Speed Calculation for Planets
The escape speed ($v_e$) from the surface of a planet is given by the formula:
$ v_e = \sqrt{\frac{2GM}{R}} $
where $G$ is the universal gravitational constant, $M$ is the mass of the planet, and $R$ is its radius.
Step-by-Step Derivation
-
Identify Given Information:
- Masses are equal: $M_1 = M_2 = M$.
- Radii are $R_1$ and $R_2$.
- Relationship: $R_2 = \frac{R_1}{2}$.
- Escape speeds are $v_1$ and $v_2$.
-
Write Escape Speed Formulas:
- For Planet $P_1$: $v_1 = \sqrt{\frac{2GM_1}{R_1}} = \sqrt{\frac{2GM}{R_1}}$.
- For Planet $P_2$: $v_2 = \sqrt{\frac{2GM_2}{R_2}} = \sqrt{\frac{2GM}{R_2}}$.
-
Calculate the Ratio $\frac{v_2}{v_1}$:
Divide the expression for $v_2$ by the expression for $v_1$:
$ \frac{v_2}{v_1} = \frac{\sqrt{\frac{2GM}{R_2}}}{\sqrt{\frac{2GM}{R_1}}} $
Simplify the ratio:
$ \frac{v_2}{v_1} = \sqrt{\frac{\frac{2GM}{R_2}}{\frac{2GM}{R_1}}} = \sqrt{\frac{R_1}{R_2}} $
-
Substitute the Radius Relationship:
Substitute $R_2 = \frac{R_1}{2}$ into the ratio:
$ \frac{v_2}{v_1} = \sqrt{\frac{R_1}{\frac{R_1}{2}}} = \sqrt{\frac{2 R_1}{R_1}} = \sqrt{2} $
The ratio $\frac{v_2}{v_1}$ is $\sqrt{2}$.
Similar Questions
The power of a crane, which lifts a mass of $1000 \text{ kg}$ to a height of $20 \text{ m}$ in $10 \text{ s}$ is :
($g = 9.8 \text{ m/s}^2$)- The magnitude and direction of the acceleration produced in a body of mass $5 \text{ kg}$ when two mutually perpendicular forces $8 \text{ N}$ and $6 \text{ N}$ act on it, are respectively :
- A box of mass $15 \text{ kg}$ is kept on the floor of a stationary trolley. The coefficient of static friction between the box and the trolley is $0.12$. Keeping the box in stationary state over the trolley, the maximum acceleration with which the trolley can be moved horizontally is : ($g = 10 \text{ m/s}^2$)
- The amount of work done to raise a mass 'm' from the surface of the Earth to a height equal to the radius of the Earth 'R', will be :
A thin wire of length 'L' and linear mass density 'm' is bent into a circular ring (in x-y plane) with centre 'C' as shown in figure. The moment of inertia of the ring about an axis yy' (tangent in the plane) will be :
- When a ruler falls vertically, 5 different persons catch it with different reaction times. ($g = 9.8 \text{ m s}^{-2}$)
A. Person A has reaction time of $0.20 \text{ s}$.
B. Person B has reaction time of $0.22 \text{ s}$.
C. Person C has reaction time of $0.18 \text{ s}$.
D. Person D has reaction time of $0.19 \text{ s}$.
E. Person E has reaction time of $0.21 \text{ s}$.
What is the correct order of the distance travelled by the ruler for each person ? - The angular speed of a flywheel is increased from $600 \text{ rpm}$ to $1200 \text{ rpm}$ in $10 \text{ s}$. The number of revolutions completed by the flywheel during this time is :
The following plots show variation of velocity ($v$) with time ($t$) of a ball thrown vertically upward, and falling back. Which of the following plots is/are correct?
A frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point $A \left( \theta = \frac{\pi}{2} \right)$ with identical uniform angular speeds in opposite directions, and meet again at point $B \left( \theta = -\frac{\pi}{2} \right)$. During this time, which of the following figures schematically represent the magnitude of the total linear momentum $\vec{P}$ of the system, as a function of $\theta$?
- A car travels on a circular racetrack of radius $50\text{ m}$, which is banked at an angle $\theta$. If the car travels at a speed $10\text{ ms}^{-1}$, then the wear and tear on its tyres is minimum. Taking the acceleration due to gravity to be $10\text{ ms}^{-2}$, the value of $\theta$ is :
Important Questions from Mechanics
A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.
Match the LIST-I with LIST-II
LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$ Choose the correct answer from the options given below:
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.
The angle of projection of a particle is measured from the vertical axis as $\phi$ and the maximum height reached by the particle is $h_m$. Here $h_m$ as function of $\phi$ can be presented as
Which of the following curves possibly represent one-dimensional motion of a particle?
