Question
What is escape velocity. Derive an expression for the same.
✓
Answer
The minimum velocity required to escape from the gravitational force of earth is called escape velocity. Total energy is the sum of P.E. and K.E.
$\text{T.E}=-\frac{\text{GMm}}{\text{R}}+\frac{1}{2}\text{mv}^2$
To escape K.E should be greater than P.E., i.e..
$\frac{1}{2}\text{mv}^2\geq-\frac{\text{GMm}}{\text{R}}$
$\text{v}_{\text{e}}=\sqrt{2\frac{\text{GM}}{\text{R}}}=\sqrt{2\text{gR}}$
$\text{T.E}=-\frac{\text{GMm}}{\text{R}}+\frac{1}{2}\text{mv}^2$
To escape K.E should be greater than P.E., i.e..
$\frac{1}{2}\text{mv}^2\geq-\frac{\text{GMm}}{\text{R}}$
$\text{v}_{\text{e}}=\sqrt{2\frac{\text{GM}}{\text{R}}}=\sqrt{2\text{gR}}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that the slope of P-V graph for an adiabatic process is $\gamma$ times that of the isothermal process.
→The potential energy function for a particle executing linear simple harmonic motion is given by $\text{V(x)} = \frac{\text{kx}^2}{2},$ where k is the force constant of the oscillator. For k = 0.5Nm-1, the graph of V(x) versus x is shown in Show that a particle of total energy 1J moving under this potential must ‘turn back’ when it reaches $\text{x} =\pm 2\text{m}.$
→What is the pressure inside the drop of mercury of radius 3.00mm at room temperature? Surface tension of mercury at that temperature (20°C) is 4.65 × 10–1N m–1. The atmospheric pressure is 1.01 × 105Pa. Also give the excess pressure inside the drop.
A small spherical ball is released from a point at a height h on a rough track shown in figure. Assuming that it does not slip anywhere, find its linear speed when it rolls on the horizontal part of the track.
One end of a spring of natural length h and spring constant k is fixed at the ground and the other is fitted with a smooth ring of mass m which is allowed to slide on a horizontal rod fixed at a height h (figure). Initially, the spring makes an angle of 37° with the vertical when the system is released from rest. Find the speed of the ring when the spring becomes vertical.
Define emissive power and absorptive power. What is the significance of their ratio?
A circular copper-ring of radius r translates in its plane with a constant velocity v. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider different pairs of diametrically opposite points on the ring.
- Between which pair of points is the emf maximum? What is the value of this maximum emf?
- Between which pair of points is the emf minimum? What is the value of this minimum emf?
Explain why:
Two bodies at different temperatures T1 and T2 if brought in thermal contact do not necessarily settle to the mean temperature (T1 + T2 )/2.
Two bodies at different temperatures T1 and T2 if brought in thermal contact do not necessarily settle to the mean temperature (T1 + T2 )/2.
Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.
Water falling from a 50m high fall is to be used for generating electric energy. If 1.8 x 105kg of water falls per hour and half the gravitational potential energy can be converted into electric energy, how many 100W lamps can be lit?