Kinetic Energy: Trial 1 vs. Trial 2

[Amik]

Homework Statement

See the picture

Relevant Equations

Conservation of momentum and energy

So, I know the right solution should be like it has same potential energy initially, in the trial one, since the floor is frictionless, the plane will move too and it shares the potential energy with the block. So in trial 1 the potential energy equals final kinetic energy of the block and plane. But in trial 2, all potential energy goes to block and thus in trial 2, the block will have greater speed
But, Can I do it this way?
In both trials, the block will have the same velocity on the plane with respect to the plane(not the ground), However, according to conservation of momentum, In trial 1, when block speed up on the plane, the plane will have horizontal velocity deirected to the left too. So the velocity of the block in trial 1 is always the horizontal component of the velocity of the block minus the horizontal component of the velocity of the plane but in trial 2 the velocity in trial 2 is always the horizontal component of the velocity of the block. So I think the velocity in trial 2 is larger.
Is there any problem with second way to do it?

 

[haruspex]

In both trials, the block will have the same velocity on the plane with respect to the plane

How so?

 

[Amik]

For plane I mean the incline. Because in both senarios blocks are only subject component of force of gravity(that force is parallel to the surface)
I might be wrong.

 

[haruspex]

For plane I mean the incline. Because in both senarios blocks are only subject component of force of gravity(that force is parallel to the surface)
I might be wrong.

The frame of reference of the (sliding) plane is non-inertial, so you need to include a pseudo force on the block to represent the plane's acceleration.

 

[Amik]

OK. This is only AP level problem. Dioes that mean my explanation is entirely wrong?

 

[haruspex]

OK. This is only AP level problem. Dioes that mean my explanation is entirely wrong?

First, the statement of yours that I quoted in post #2 is not obviously true so you cannot use that argument without more detailed reasoning.
Your explanation in post #3 leans on use of a non-inertial frame, so needs to be particularly careful. In particular, you have omitted the pseudo force that needs to be included in such a frame.
Your explanation might still work if you include the pseudo force and develop the relevant equations.