PS.7.1 Worked Example - Collision and Sliding on a Rough Surface

A diagram of two colliding blocks.

Block 1 of mass \(\displaystyle 3m\) is sliding along a frictionless horizontal table to the right with speed \(\displaystyle v_0\). Block 1 collides with block 2 of mass \(\displaystyle m\) that is moving to the left with speed \(\displaystyle v_0\). After the collision, the two blocks stick together and the blocks enter a rough surface at \(\displaystyle x=0\) with a coefficient of kinetic friction that increases with distance as \(\displaystyle \mu (x)=b x^2\) for \(\displaystyle 0\leq x\leq d\), where \(\displaystyle b\) is a positive constant. The blocks come to rest at \(\displaystyle x=d\). The downward gravitational acceleration has magnitude \(\displaystyle g\). Determine an expression for the initial speed \(\displaystyle v_0\) of the blocks.


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