Answer to Question #99635 in Mechanics | Relativity for James

Question

Let's say that a small block of mass is released from rest at the top on a frictionless wedge. The wedge is also on a frictionless surface. We know the values of the masses and the final velocity of the small block. Due to the conservation of momentum, the wedge moves in the opposite direction (P=0 before and there is no external force) and we can find the velocity of the wedge at the end. The question is to find the height of the wedge (the height at which the block was released).
1-Do we need to take the total final kinetic energy including the kinetic energy of the wedge to find the initial potential energy or only the final k.e. of the block that slid down?
2-How exactly does the wedge gain energy?

Answer 1

Due to the absence of friction, the mechanical energy of the wedge-block system is conserved.

1- The kinetic energy of the wedge must be taken into account. The law of conservation of energy is:

$\frac{mv^2}{2}+\frac{MV^2}{2}=mgh$

On the left side of the equation is the sum of the kinetic energy of the block and the wedge. On the right, the potential energy of the block is recorded before it begins to move.

Due to the conservation of momentum we have the second equation

$mv_x+MV=0$

The sum of the pulses of the block and the wedge at the end of the wedge is 0 because at the beginning of the process both bodies were at rest. From the second equation, we determine the speed of the wedge, which has only one component - horizontal.

$V=-\frac{mv_x}{M}$

We see that the wedge moves in the direction opposite to the horizontal component of the block velocity. Substituting this expression into the first equation and making simple transformations, we obtain the value of the initial block height.

Answer: $h=\frac{v^2+(m/M)v_x^2}{2g}$

2- According to Newton’s third law, the forces of interaction of two material points are equal in magnitude, oppositely directed, and act along the straight line connecting these material points. The figure shows the forces acting in the system (not all). Since there is no friction force, the support reaction force $N_b$ is strictly perpendicular to the upper surface of the wedge. The force acting from the block on the wedge $N_w$ is equal $N_b$ in magnitude and directed in the opposite direction. The vertical component of this force $N_{wy}$ is compensated by the surface and the horizontal one $N_{wx}$ sets the wedge in motion. The work of this force component increase the kinetic energy of the wedge.