Question

Q) Two masses m1 and m2 are released from rest to move towards each other under mutual gravitational attraction . Initially they were large distance apart . What is their relative speed when they are a distance 'r' apart ?

Q) Write Kepler's law .

Answer 1

Explanation:

Kepler's three laws of planetary motion can be stated as follows: (1) All planets move about the Sun in elliptical orbits, having the Sun as one of the foci. (2) A radius vector joining any planet to the Sun sweeps out equal areas in equal lengths of time.

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Answer 2

${\huge{\underline{\underline{\rm{Question:-}}}}}$

Q) Two Masses $\sf m_1$ and $\sf m_2$ are released from rest to move towards each other under mutual Gravitational attraction . Initially they were large distance apart . What is their relative speed when they are r distance apart ?

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As , no other forces are present , leaving aside the internal Gravitational attraction , the Total mechanical energy and linear momentum of system remains conserved .

So , We can write

$\rm \sum {p _{i}} = \sum{p _{f}}$

$\mapsto \rm \: 0 = m _{1}v _{1} - m _{2}v _ {2} \: \: \: \: \: \: \: \: ....(i)$

Negative sign has been chosen to keep v1 and v2 as magnitude of velocities only .

Also ,

$\rm U_i + K_i = U_f + K_f$

${\rm{\mapsto\;0+0=\dfrac{-Gm_1m_2}{r}+\dfrac{1}{2}m_1{v_1}^2+\dfrac{1}{2}m_2{v_1}^2\;\;\;\;....(ii)}}$

From (i) , we get

$\mapsto \rm \: m _{1}v _{1} = m _{2}v _{2} = p \: (say)$

From (ii) , we get

$\rm\dfrac{-Gm_1m_2}{r} = \dfrac{p^2}{2m_1}+\dfrac{p^2}{2m_2}\;\;\{as\;k=\dfrac{p^2}{2m}\}$

$\mapsto\rm p=m_1m_2\sqrt{\dfrac{2G}{(m_1m_2)r}}$

Now ,

Relative speed = $\rm v_1+v_2$

$= \rm \: \frac{p}{m _{1}} + \frac{p}{m _{2}}$

$\underline{ \boxed{ \rm{ \purple{Relative \: Speed = \sqrt{ \frac{2G(m _{1} m _{2})}{r} } }}}}$

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${\large{\rm{\underline{\star\; Kepler's\;Law:-}}}}$

# Kepler's First Law :

All planets move in elliptical orbits , with the sun at one foci of the ellipse .

# Kepler's Second Law :

The line that joins any planet to the sun sweeps out equal area in equal intervals of time .

# Kepler's Third Law :

The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet .

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