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
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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 $v_2$.
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Write Escape Speed Formulas:
- For Planet $P_1$: $v_1 = \sqrt{\frac{2GM_1}{R_1}} = \sqrt{\frac{2GM}{R_1}}$.
- For Planet $P_2$: $v_2 = \sqrt{\frac{2GM_2}{R_2}} = \sqrt{\frac{2GM}{R_2}}$.
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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}} $
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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 :
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- 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 :
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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 :
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Important Questions from Mechanics
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
Suppose there is a uniform circular disc of mass $M \text{ kg}$ and radius $r \text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^2$. The value of $x$ is ________.
- A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t = A \rho^a r^b \eta^c \sigma^d$, where $A$ is a constant and $a$, $b$, $c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is ________.
