The mass and radius of the earth and moon are | vedclass

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

Question

(A) The gravitational potential energy $U$ of a particle of mass $m_0$ at the midpoint between the earth and the moon is given by the sum of potential

Vedclass · Accepted Answer

$2 \sqrt{\frac{G}{d}(M+m)}$

Vedclass · Answer

(A) The gravitational potential energy $U$ of a particle of mass $m_0$ at the midpoint between the earth and the moon is given by the sum of potentials due to both bodies:$U = -\frac{G M m_0}{d/2} - \frac{G m m_0}{d/2} = -\frac{2 G m_0}{d} (M + m)$To reach infinity,the total energy of the particle must be at least zero. Let $V$ be the minimum projection velocity.Applying the principle of conservation of energy:$K_i + U_i = K_f + U_f$$\frac{1}{2} m_0 V^2 - \frac{2 G m_0}{d} (M + m) = 0 + 0$$\frac{1}{2} m_0 V^2 = \frac{2 G m_0}{d} (M + m)$$V^2 = \frac{4 G}{d} (M + m)$$V = 2 \sqrt{\frac{G(M+m)}{d}}$

The mass and radius of the earth and moon are $M, R$ and $m, r$ respectively. The distance between their centers is $d$. The minimum velocity with which a particle of mass $m_0$ should be projected from the midpoint between them so that it could reach infinity is

Similar Questions

$A$ mass of $6 \times 10^{24} \ kg$ is to be compressed into a sphere in such a way that the escape velocity from the sphere is $3 \times 10^8 \ m/s$. What should be the radius of the sphere? (Given: $G = 6.67 \times 10^{-11} \ N \cdot m^2/kg^2$)

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

The escape velocity of an object on a planet whose $g$ value is $9$ times that of Earth and whose radius is $4$ times that of Earth in $km/s$ is:

$A$ rocket is launched with velocity $10 \ km/s$. If the radius of the Earth is $R$,then the maximum height attained by it will be: (in $R$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module