A meteor of mass m having a speed v at infini | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) According to the law of conservation of energy,the total energy of the meteor at infinity is equal to the total energy at the earth's surface.At i

Vedclass · Accepted Answer

$\sqrt{v^2 + v_e^2}$

Vedclass · Answer

(A) According to the law of conservation of energy,the total energy of the meteor at infinity is equal to the total energy at the earth's surface.At infinity,the potential energy is $0$ and the kinetic energy is $\frac{1}{2}mv^2$.At the earth's surface,the potential energy is $-\frac{GMm}{R}$ and the kinetic energy is $\frac{1}{2}mv_f^2$,where $v_f$ is the final speed.So,$\frac{1}{2}mv^2 + 0 = \frac{1}{2}mv_f^2 - \frac{GMm}{R}$.We know that the escape speed $v_e = \sqrt{\frac{2GM}{R}}$,which implies $v_e^2 = \frac{2GM}{R}$,or $\frac{GM}{R} = \frac{v_e^2}{2}$.Substituting this into the energy equation: $\frac{1}{2}mv^2 = \frac{1}{2}mv_f^2 - m(\frac{v_e^2}{2})$.Dividing by $\frac{m}{2}$ gives $v^2 = v_f^2 - v_e^2$.Therefore,$v_f^2 = v^2 + v_e^2$,which means $v_f = \sqrt{v^2 + v_e^2}$.

Column-$I$	Column-$II$
$(1)$ Value of escape velocity on the surface of the Earth	$(a)$ $2.38 \, km \, s^{-1}$
$(2)$ Value of escape velocity on the surface of the Moon	$(b)$ $7.92 \, km \, s^{-1}$
	$(c)$ $11.2 \, km \, s^{-1}$

$A$ meteor of mass $m$ having a speed $v$ at infinity reaches the surface of the earth with a speed of (where $v_e$ is the escape speed from the earth's surface).

Similar Questions

$A$ body is thrown with a velocity equal to $n$ times the escape velocity $(v_e)$. What will be the velocity of the body at a large distance away?

The mass of a planet is half that of the Earth and the radius of the planet is one-fourth that of the Earth. If we plan to send an artificial satellite from the planet,the escape velocity will be (escape velocity on Earth $v_e = 11 \ km \ s^{-1}$): (in $km \ s^{-1}$)

Match Column-$I$ with Column-$II$.

Column-$I$ Column-$II$

$(1)$ Value of escape velocity on the surface of the Earth $(a)$ $2.38 \, km \, s^{-1}$

$(2)$ Value of escape velocity on the surface of the Moon $(b)$ $7.92 \, km \, s^{-1}$

$(c)$ $11.2 \, km \, s^{-1}$

The escape velocity of a planet having mass $6$ times and radius $2$ times as that of Earth is

$A$ body is projected vertically from the surface of the earth of radius $R$ with a velocity equal to half of the escape velocity. The maximum height reached by the body is

Vedclass Test Series

Exam Paper Generator

Online Exam Module