Gravitational Potential Energy of a Two Mass System

Q. A particle of mass $m$ is projected with a velocity $v=k{v}_e(k<1)$ from the surface of the earth. $(v_e=$ escape velocity$)$. The maximum height above the surface reached by the particle is -

$R$ is the radius of earth.

1. $R{\left(\frac{k}{1-k}\right)}^2$
2. $R{\left(\frac{k}{1+k}\right)}^2$
3. $\frac{R^{2}k}{1+k}$
4. $\frac{R{k}^2}{1-k^2}$

Q. A particle is fired vertically upwards with a speed of $9.8km/s$. Find the maximum height attained by the particle. Radius of earth = $6400km$ and $g$ at the surface = $9.8{m/s}^2$. Consider only earth’s gravitation.

1. $50100km$
2. $20900km$
3. $43200km$
4. $35200km$

Q. A rocket of mass M is launched vertically from the surface of the earth with an initial speed V. Assuming the radius of the earth to be R and negligible air resistance, the maximum height attained by the rocket above the surface of the earth is

1. $\frac{R}{(\frac{gR}{2V^2}-1)}$
2. $R(\frac{gR}{2V^2}-1)$
3. $\frac{R}{(\frac{2gR}{V^2}-1)}$
4. $R(\frac{2gR}{V^2}-1)$

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is

1. ${\left[2G\frac{(m_1-m_2)}{r}\right]}^{1/2}$
2. ${\left[\frac{2G}{r}\right(m_1+m_2\left)\right]}^{1/2}$
3. ${\left[\frac{r}{2G\left(m_1{m}_2\right)}\right]}^{1/2}$
4. ${\left[\frac{2G}{r}{m}_1{m}_2\right]}^{1/2}$

Q. The minimum energy required to launch a $m\text{kg}$ satellite from the earth’s surface in a circular orbit at an altitude $2R$, where $R$ is the radius of earth is

1. $\frac{5}{3}mgR$
2. $\frac{4}{3}mgR$
3. $\frac{5}{6}mgR$
4. $\frac{5}{4}mgR$

Q. A space station is set up in space at a distance equal to the earth’s radius from the surface of the earth. Suppose a satellite can be launched from the space station. Let $v_1$ and $v_2$ be the escape velocities if the satellite on the earth’s surface and space station, respectively. Then,

1. $v_2=v_1$
2. $v_2<v_1$
3. $v_2>v_1$
4. (a), (b) and (c) are valid depending on satellite.

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at an infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

1. ${\left[2G\frac{(m_1-m_2)}{r}\right]}^{1/2}$
2. ${\left[\frac{2G}{r}\right(m_1+m_2\left)\right]}^{1/2}$
3. ${\left[\frac{r}{2G\left(m_1{m}_2\right)}\right]}^{1/2}$
4. ${\left[\frac{2G}{r}{m}_1{m}_2\right]}^{1/2}$