Problem 3 - Disc on moving block with curved track

Question:

A body of mass $M$ (see figure 1.43) with a small disc of mass $m$ placed on it rests on a smooth horizontal plane. The disc is set in motion in the horizontal direction with velocity $v$. To what height (relative to the initial level) will the disc rise after breaking off the body M? The friction is assumed to be absent.

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Solution

When $m$ is just about to break off form $M$, they no longer exert a normal force on one another and should therefore be moving with the same velocity along the x direction.

Using conservation of momentum in the x axis:

$$mv=(m+M)v_x$$

Therefore

$$v_x=\frac{mv}{m+M}$$

And from conservation of energy at the point when $m$ breaks from $M$

$$\frac{1}{2}m{v}^2=\frac{1}{2}(m+M)v_x^2+mg{h}_1+\frac{1}{2}m{v}_y^2$$

Conserving energy after $m$ breaks from $M$

$$\frac{1}{2}m{v}_y^2=mg{h}_2$$

Combining all equations:

$$\frac{1}{2}m{v}^2=\frac{1}{2}(m+M)(\frac{mv}{m+M}{)}^2+mg(h_1+h_2)$$

Rearranging, we get:

$$h_1+h_2=\frac{M{v}^2}{2g(m+M)}$$