Escape Speed

Q. Escape velocity of a $1\text{kg}$ body on a planet is $100\text{m/s}$. Potential energy of body at that planet is :

1. $-2400\text{J}$
2. $-5000\text{J}$
3. $-1000\text{J}$
4. $-10000\text{J}$

Q. The escape velocity for a planet is $v_e$. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the center of the planet, its speed will be

1. $v_e$
2. $\frac{v_e}{\sqrt{2}}$
3. $\frac{v_e}{2}$
4. Zero

Q. A particle is projected vertically upward from the surface of the earth with a speed of $\sqrt{\frac{3}{2}gR}$, $R$ being the radius of the earth and g is the acceleration due to gravity on the surface of the earth. Then the maximum height ascended is (neglect cosmic dust resistance)

1. $\frac{3R}{4}$
2. $R$
3. $2R$
4. $3R$

Q. An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth. If the satellite is stopped suddenly in its orbit and allowed to fall freely onto the earth, find the speed $\text{km/s}$ with which it hits surface of the earth. $(g=10{\text{m/s}}^2\text{and}{R}_E=6400\text{km})$

1. $4$
2. $10$
3. $6$
4. $8$

Q. Two satellites of masses $400\text{kg}$ and $500\text{kg}$ are revolving around earth in different circular orbits of radii $r_1$ and $r_2$ such that their kinetic energies are equal. The ratio of $r_1$ to $r_2$ is

1. $16:25$
2. $5:4$
3. $25:16$
4. $4:5$

Q. The distance of a planet from the sun is $5$ times the distance between the earth and the sun. The Time period of the planet is:

1. $5^{\frac{3}{2}}years$
2. $5^{\frac{2}{3}}years$
3. $5^{\frac{1}{3}}years$
4. $5^{\frac{1}{2}}years$

Q. Pertaining to two planets, the ratio of escape velocities from respective surfaces is $1:2$ , the ratio of the time period of the same simple pendulum at their respective surfaces is $2:1$ (in same order). Then the ratio of their average densities is

1. $1:1$
2. $1:2$
3. $1:4$
4. $8:1$

Q. A planet in a distant solar system is $10$ times more massive than the earth and its radius is $10$ times smaller. Given that the escape velocity from the earth is $11\text{km/s}$, the escape velocity from the surface of the planet would be

1. $1.1\text{km/s}$
2. $11\text{km/s}$
3. $110\text{km/s}$
4. $0.11\text{km/s}$

Q. A shell of mass $M$ and radius $R$ has a point mass $m$ placed at a distance $r$ from its centre. The gravitational potential energy $U\left(r\right)$ vs $r$ will be

1. 
2. 
3. 
4. 

Q. Escape velocity from earth is about $11km{s}^{-1}$. Escape velocity from a planet having twice the radius and same mean density as the earth is.

1. $22km{s}^{-1}$
2. $15.5km{s}^{-1}$
3. $5.5km{s}^{-1}$
4. $11km{s}^{-1}$

Q. A planet has radius equal to $2R_E$ and density is equal to $4{\rho }_E$. If the value of escape speed at surface of planet is $\alpha$ times $v_E$ $(v_E$ is escape speed on earth, $R_E$ is radius of earth and ${\rho }_E$ is the density of earth$)$ then -

1. $\alpha =2$
2. $\alpha =9$
3. $\alpha =6$
4. $\alpha =4$

Q.

A space-ship is put into a circular orbit close to Earth's surface. What additional velocity must be imparted to the ship so that it is able to escape the gravitational pull of Earth(approximately)? $(R=6400km,g=9.8\frac{m}{s^2})$

1. $11.2\frac{km}{s}$

2. $3.3\frac{km}{s}$

3. $6.8\frac{km}{s}$

4. $9.3\frac{km}{s}$

Q. A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass $=5.98\times {10}^{24}\text{kg}$) have to be compressed to be a black hole?

1. ${10}^{-9}\text{m}$
2. ${10}^{-6}\text{m}$
3. ${10}^{-2}\text{m}$
4. $100\text{m}$

Q. A planet has radius equal to $2R_E$ and density is equal to $4{\rho }_E$. If the value of escape speed at surface of planet is $\alpha$ times $v_E$ $(v_E$ is escape speed on earth, $R_E$ is radius of earth and ${\rho }_E$ is the density of earth$)$ then -

1. $\alpha =2$
2. $\alpha =9$
3. $\alpha =6$
4. $\alpha =4$

Q. A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass $=5.98\times {10}^{24}\text{kg}$) have to be compressed to be a black hole?

1. ${10}^{-9}\text{m}$
2. ${10}^{-6}\text{m}$
3. ${10}^{-2}\text{m}$
4. $100\text{m}$

Q.

Escape velocity of a body on a planet is $100m{s}^{-1}$. What is the gravitational potential energy of the body at the surface, if the mass of the body is $1kg$?

1. $-5000J$

2. $-1000J$

3. $-2400J$

4. $5000J$

Q. A projectile is projected with velocity $k{v}_e$ in vertically upward direction from the ground into the space.($v_e$ is escape velocity and k<1). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go, will be : (R = radius of earth)

1. $\frac{R}{k^2+1}$
2. $\frac{R}{k^2-1}$
3. $\frac{R}{1-k^2}$
4. $\frac{R}{k+1}$

Q. The escape speed of any projectile on Earth's surface is 11.2 km/s. A body is projected with thrice this speed. What is the speed of body far away from earth? Ignore presence of Sun & other planets.

1. 31 km/sec
2. 33.6 km/sec
3. 27.8 km/sec
4. 22.8 km/sec

Q.

The height at which g will be $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ th of its value at the earths surface is

1. $h=R$

2. $h=\frac{R}{2}$

3. $h=2R$

4. none of these

Q. A planet has radius equal to $2R_E$ and density is equal to $4{\rho }_E$. If the value of escape speed at surface of planet is $\alpha$ times $v_E$ $(v_E$ is escape speed on earth, $R_E$ is radius of earth and ${\rho }_E$ is the density of earth$)$ then -

1. $\alpha =2$
2. $\alpha =9$
3. $\alpha =6$
4. $\alpha =4$

Q. A particle is projected vertically upward from the surface of the earth with a speed of $\sqrt{\frac{3}{2}gR}$, $R$ being the radius of the earth and g is the acceleration due to gravity on the surface of the earth. Then the maximum height ascended is (neglect cosmic dust resistance)

1. $\frac{3R}{4}$
2. $R$
3. $2R$
4. $3R$

Q. A projectile is projected with velocity $k{v}_e$ in vertically upward direction from the ground into the space.($v_e$ is escape velocity and k<1). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go, will be : (R = radius of earth)

1. $\frac{R}{k^2+1}$
2. $\frac{R}{k^2-1}$
3. $\frac{R}{1-k^2}$
4. $\frac{R}{k+1}$