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Please assume that there is no friction involved. In any wedge constraints problem,for example if two blocks are sliding over each other like this-

enter image description here

Sorry for the gap between both of them, they are actually in contact. Block B will slide down and block A will move in forward direction.
Now, I have often seen people finding the constraint-relation between velocities of the two blocks by making the velocity component of A and B along the normal to be equal.

What I wanted to know is that why take the velocity component along the normal to be equal,why not any other velocity component of the two blocks.

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In your sketch, the horizontal velocity for block B assumes it is moving on a horizontal surface. For block A to have a vertical velocity, there must be a horizontal force that prevents it from moving to the left. (The normal force has a horizontal component.) The forces must be equal and opposite at the contact surface. The only constraint on velocities is that the two surfaces must remain in contact.

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Short answer: It has not so much to do with the 'component along the normal', as it is with the component of velocity 'perpendicular to the surface'.

Imagine any two blocks constrained to maintain contact on one of their surfaces. What should be the minimum necessary condition regarding the velocities of these blocks required to persist this kind of motion? Think.

(HINT: It has something to do with the separation between the contact surfaces of the two blocks.)

Obviously, the first thing that comes to mind is that the separation between the surfaces cannot change, no matter what. It has to remain at zero, or constrained motion will no longer persist; and for the separation between the two surfaces to remain at zero, their relative displacements must be zero as well. As logic follows, their relative velocities must also be zero at all times. This is exactly what we do when we equate their velocities perpendicular to the surface: $(\vec v_A(_\bot))-(\vec v_B(_\bot))=0$

Therefore it simply makes intuitive sense to take the components of velocities perpendicular to the surfaces, which also happens to be the direction along which the normal forces act.

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  • $\begingroup$ With due respect,I didn't get the last paragraph you wrote.I mean why the relative velocity perpendicular to the contact surface between the two bodies must be 0 ,why can't we take any other relative velocities to be 0. Like the velocity of A which is completely in downward direction,this is also helping A to stick with B right??? $\endgroup$
    – PATRICK
    Commented Aug 31, 2020 at 18:28
  • $\begingroup$ It's true that in any one particular scenario, there is not just one direction in which relative velocities are zero. Take any example where constrained motion between the surfaces of two blocks is maintained, you will see that there are infinite components of relative velocity that will be zero(except the parallel ones in the cases where the blocks are slipping) provided that the surfaces are flat, as if two points don't have any perpendicular velocities relative to each other then they are also not moving diagonally away from each other. You can apply the same logic in this scenario too. $\endgroup$
    – Reet Jaiswal
    Commented Aug 31, 2020 at 18:43
  • $\begingroup$ What I've mentioned is a fundamental generalisation that can be applied in all scenarios, and because it's easier to find components along the normal. $\endgroup$
    – Reet Jaiswal
    Commented Aug 31, 2020 at 18:44

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