A particle released at a large distance from | vedclass

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

Question

(A) Let the mass of the particle be $m$,the mass of the planet be $M$,and its radius be $R$. Taking the gravitational potential at a large distance (i

Vedclass · Accepted Answer

$\sqrt{1.5} v_e$

Vedclass · Answer

(A) Let the mass of the particle be $m$,the mass of the planet be $M$,and its radius be $R$. Taking the gravitational potential at a large distance (infinity) as zero,the total energy of the particle at infinity is $E_i = 0$. The gravitational potential at the centre of the planet is $V_c = -\frac{3GM}{2R}$. By the law of conservation of energy,the total energy at infinity equals the total energy at the centre of the planet: $K_i + U_i = K_c + U_c$ $0 + 0 = \frac{1}{2}mv^2 + m(-\frac{3GM}{2R})$ $\frac{1}{2}mv^2 = \frac{3GMm}{2R}$ $v^2 = \frac{3GM}{R}$ $v = \sqrt{\frac{3GM}{R}}$. We know that the escape velocity at the surface of the planet is $v_e = \sqrt{\frac{2GM}{R}}$,which implies $\frac{GM}{R} = \frac{v_e^2}{2}$. Substituting this into the expression for $v$: $v = \sqrt{3 \cdot \frac{v_e^2}{2}} = \sqrt{1.5} v_e$.

$A$ particle released at a large distance from a planet reaches the planet only under gravitational attraction and passes through a smooth tunnel through its centre. If $v_e$ is the escape velocity of a body at the planet,then the particle's speed at the centre of the planet is

Similar Questions

The Earth is assumed to be a sphere of radius $R$. $A$ platform is arranged at a height $R$ from the surface of the Earth. The escape velocity of a body from this platform is $fv$,where $v$ is its escape velocity from the surface of the Earth. The value of $f$ is

Let the escape velocity of a body kept at the surface of a planet be $u$. If it is projected at a speed of $200 \%$ more than the escape speed,then its speed in interstellar space will be ...........

There is no atmosphere on the moon because

An object is kept at rest at a distance of $3R$ above the earth's surface,where $R$ is the earth's radius. The minimum speed with which it must be projected so that it does not return to earth is (Assume $M =$ mass of earth,$G =$ Universal gravitational constant).

Escape velocity at the surface of the Earth is $11.2 \, km/s$. If the radius of a planet is double that of the Earth but its mean density is the same as that of the Earth,then the escape velocity will be ........ $km/s$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module