Newton's law of gravitation, energy and centre of mass question

[mrpurpletoes]

Homework Statement

two point sized objects of mass 2kg and 10kg are at rest when kept at an infinite distance from each other. at some time, the velocity of the 10kg object is 2 m/s towards the 2kg object, find the velocity of the 2kg object at that time.

Relevant Equations

F=GmM/r^2

our professor told us to solve this using the fact that the velocity of centre of mass (com) will be 0, however before he said this i had taken a different approach and did this:
ma=GmM/r^2
m(dv/dr)*(dr/dt)= Fg (taking GmM/r^2 as Fg for now)
1/2(mv^2)= Fg dr
and since for M force= -Fg
1/2(Mv^2)=-Fg dr 1/2(mv^2)+ 1/2(Mv^2)=0, however, using this i get a different answer than when using the velocity of com=0, could someone explain please?

 

[PeroK]

I'm not sure what your approach is. Are you proposing to do some integration?

Using a conservation law (I would have used conservation of momentum in this case) avoids having to integrate.

 

[mrpurpletoes]

I'm not sure what your approach is. Are you proposing to do some integration?

Using a conservation law (I would have used conservation of momentum in this case) avoids having to integrate.

yes, our professor said to use conservation of momentum/ conservation of velocity of com, and i understood the method and the reasoning behind it, the part im facing a problem with is, why upon using the method i used am i getting a different answer.

 

[PeroK]

yes, our professor said to use conservation of momentum/ conservation of velocity of com, and i understood the method and the reasoning behind it, the part im facing a problem with is, why upon using the method i used am i getting a different answer.

I don't understand your method.

 

[mrpurpletoes]

sorry ill try to make it clearer, (Fg=GmM/x²) (assuming velocity of M to be V and velocity of m to be v at a point)
for the mass M
Ma= Fg
M(dV/dt)= Fg
M(dV/dx)*(dx/dt)=Fg
M(dV/dx)*V=Fg
MVdV=Fg*dx
MV²/2=Fg*dx .....(1)

for the mass m
m(-a)= -Fg
m(dv/dt)= Fg
m(dv/dx)*(dx/dt)=Fg
m(dv/dx)*v=Fg
mvdv=Fg*dx
mv²/2=Fg*dx ......(2)

so from (1) and (2)
mv²/2 should be equal to MV²/2, however when i use these two equations to find the solution i am getting a different answer than when i use conservation of momentum.

 

[PeroK]

sorry ill try to make it clearer, (Fg=GmM/x²) (assuming velocity of M to be V and velocity of m to be v at a point)
for the mass M
Ma= Fg
M(dV/dt)= Fg
M(dV/dx)*(dx/dt)=Fg
M(dV/dx)*V=Fg
MVdV=Fg*dx
MV²/2=Fg*dx .....(1)

for the mass m
m(-a)= -Fg
m(dv/dt)= Fg
m(dv/dx)*(dx/dt)=Fg
m(dv/dx)*v=Fg
mvdv=Fg*dx
mv²/2=Fg*dx ......(2)

so from (1) and (2)
mv²/2 should be equal to MV²/2, however when i use these two equations to find the solution i am getting a different answer than when i use conservation of momentum.

Kinetic energy depends on a force acting over a distance. Not over time. The $dx$ in your two cases are different. The large mass accelerates more slowly than the small mass, so $dx_M/dt$ will be less than $dx_m/dt$.

Momentum depends on a force acting over a time. If the force has the same magnitude on each mass, then the magnitude of the momentum of each mass stays the same.

 

[mrpurpletoes]

thanks a lot dude o7