2 object gravitation, collision speed

AI Thread Summary
To determine the collision speed of two objects with mass m and radius r, initially at rest and a distance L apart, one must apply Newton's law of gravitation, Fg = G m^2/(r^2). The discussion emphasizes the need to formulate differential equations to describe the motion of the objects as they accelerate towards each other. A suggested approach involves using the relationship a = dv/dx * v to derive the necessary equations. The poster expresses uncertainty in writing these equations despite having solved a similar problem previously. Clarification on the formulation of the equations is sought to progress in finding the velocity at the moment of collision.
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Homework Statement


2 objects each with mass, m, and radius, r, are unaffected by any other objects. They are the distance L apart. At time t=0 , both are at rest. From then on they accelerate towards each other.
Determine the speed of the planets at the moment of collision.

Homework Equations


From assignment: "Use a= dv/dx * v in order to solve the problem".
Newtons law of gravitation: Fg= G m^2/(r^2)

The Attempt at a Solution



Well I would think that I'm supposed to make 1 or 2 differential equations, which I can then solve specifically by x_0=0 .
My problem is writing the equations, can anyone help, please?
 
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I have solved a very similar problem, but am uncertain of the solution! I obtained a position-time equation, from which I could easily differentiate and find velocity... Umm refer to my post in general physics if you want to see that.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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