Two bodies of masses m1 and m2 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

Question

Two bodies of masses m1 and m2 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 (A) [2 G ((m1-m2)/r)]1 / 2. (B) [(2 G/r)(m1+m2)]1 / 2 (C) [(r/2 G(m1 m2))]1 / 2 (D) [(2 G/r) m1 m2]1 / 2

Tardigrade Admin · Accepted Answer

Apply the principle of conservation of momentum and conservation of energy.
Let velocities of these masses at

r

distance from each other be

v_{1}

and

v_{2}

respectively.
By conservation of momentum

m_{1} v_{1} - m_{2} v_{2} = 0

\Rightarrow m_{1} v_{1} = m_{2} v_{2}

(i)
By conservation of energy
Change in

PE =

change in

K E

\frac{G m _{1} m _{2}}{r} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}

\Rightarrow \frac{m _{1}^{2} v _{1}^{2}}{m _{1}} + \frac{m _{2}^{2} v _{2}^{2}}{m _{2}} = \frac{2 G m _{1} m _{2}}{r}

(ii)
On solving Eqs. (i) and (ii)

v_{1} = \frac{2 G m _{2}^{2}}{r ( m _{1} + m _{2} )} and v_{2} = \frac{2 G m _{1}^{2}}{r ( m _{1} + m _{2} )}

∴ v_{a pp} = ∣ v_{1} ∣ + ∣ v_{2} ∣ = \frac{2 G}{r} (m_{1} + m_{2})

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

$[2 G \frac{( m _{1} - m _{2} )}{r}]^{1/2}$ .

$[\frac{2 G}{r} (m_{1} + m_{2})]^{1/2}$

$[\frac{r}{2 G ( m _{1} m _{2} )}]^{1/2}$

$[\frac{2 G}{r} m_{1} m_{2}]^{1/2}$

Q. Two bodies of masses m1​ and m2​ 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

[2Gr(m1​−m2​)​]1/2.

[r2G​(m1​+m2​)]1/2

[2G(m1​m2​)r​]1/2

[r2G​m1​m2​]1/2

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

$[2 G \frac{( m _{1} - m _{2} )}{r}]^{1/2}$ .

$[\frac{2 G}{r} (m_{1} + m_{2})]^{1/2}$

$[\frac{r}{2 G ( m _{1} m _{2} )}]^{1/2}$

$[\frac{2 G}{r} m_{1} m_{2}]^{1/2}$