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202210251204 Dynamics Figures (Elementary) Q7

126. Two carts are pushed apart by an explosion of a powder charge Q placed between them. The cart weighing 100\,\mathrm{kg} travels a distance of 18 metres and stops. What distance will be covered by the other cart weighing 300\,\mathrm{kg}? The coefficients of friction k between the ground and the carts are the same.

Modified from B. Bukhovtsev et al. (1978). Problems In Elementary Physics.

First, have a picture in mind:

Applying the law of conservation of linear momentum at the instant of impact,

\begin{aligned} m_A\mathbf{u}_A+m_B\mathbf{u}_B & = (m_A+m_B)\mathbf{0} \\ -100u_A+300u_B & = 0 \\ u_A & = 3u_B \\ \end{aligned}

and the law of conservation of mechanical energy after then,

\begin{aligned} \textrm{KE}_A & = W_{f_A} \\ \frac{1}{2}m_Au_A^2 & = km_Ags_A \\ u_A & = 6\sqrt{kg} \\ u_B & = 2\sqrt{kg} \\ \cdots\cdots\cdots\cdots\enspace & \enspace\cdots\cdots\cdots\cdots \\ \textrm{KE}_B & = W_{f_B} \\ \frac{1}{2}m_Bu_B^2 & = km_Bgs_B \\ \frac{1}{2}(300)(2\sqrt{kg})^2 & = k(300)gs_B \\ s_B & = 2\\ \end{aligned}

\therefore The other cart will travel a distance of 2\,\mathrm{m}.