Why in two block systems do both blocks tend to have the same velocity?

[harsh_23]

TL;DR Summary

Why in two block system, both block tends to have same velocity at maximum potential energy when no external force is applied?
Like how can I prove it and why is it the case?

So giving two examples where this happens

Example 1
Consider a hemispherical bowl and a block placed inside it if the bowl is given velocity v𝑣 and given that there is no friction, the block will gain kinetic energy and then will rise to a maximum height h both the bowl and block will have same velocity in horizontal direction and both will have 0 velocity in vertical direction

I get that in vertical direction both will have 0 velocity but why both the block will have same velocity at topmost point.

Example 2
take a two block connected by spring both block are given some velocity then maximum compression or expansion in spring happens when both the blocks have same velocity , here we can get intuition as when both block reaches same velocity there's no chance of maximum compression or expansion but how can I prove it?

 

[Orodruin]

Consider the center of mass frame. In that frame both masses will have equal and opposite momentum. The kinetic energy is minimised when that momentum is zero, ie, when both objects are at rest in the center of mass frame.

 

[harsh_23]

Consider the center of mass frame. In that frame both masses will have equal and opposite momentum. The kinetic energy is minimised when that momentum is zero, ie, when both objects are at rest in the center of mass frame.

So in cm frame all the kinetic energy is converted to potential energy , then why does in ground frame all the kinetic energy is not converted to potential energy of the system, are there any restriction? Or is it to do with the system as block leaves the bowl and not net horizontal force in x direction , ok I seem to understand things , so it is better to ask why all the kinetic energy becomes zero in cm frame ?

 

[jbriggs444]

So in cm frame all the kinetic energy is converted to potential energy , then why does in ground frame all the kinetic energy is not converted to potential energy of the system, are there any restriction?

Conservation of momentum imposes a restriction. The center of mass always moves at a velocity of $v_\text{cm} = \frac{p}{m_\text{tot}}$. Even if all the pieces are motionless relative to each other, the system as a whole must have a minimum kinetic energy of $KE_\text{min} = \frac{1}{2}m_\text{tot}{v_\text{cm}}^2$

 

[A.T.]

it is better to ask why all the kinetic energy becomes zero in cm frame ?

Because in the CM frame the momenta of the bodies are equal but opposite, and it's a periodic system so they have to swap signs at some time point, which is when both bodies are at rest.

 

[harsh_23]

One more thing is center of mass frame is still zero momentum frame when it is non inertial?

Like I can't conserve momentum when is is non inertial.

 

[A.T.]

One more thing is center of mass frame is still zero momentum frame when it is non inertial?

Yes.

Like I can't conserve momentum when is is non inertial.

Not in general, but in this specific non-inertial frame momentum is conserved.