Randall D. Knight — Newton's Third Law: Problem Challenge_Problem_59

FIGURE CP7.59 shows a small block of mass \( m \) sliding with velocity \( v_x \) across a very long plate of mass \( M \). The plate itself is sliding across a frictionless surface with velocity \( V_x \). The plate has a viscous upper surface, so the retarding force on the block is not the usual kinetic friction but, instead, a linear drag force \((F_{\text{drag}})_x = -b v_{\text{rel}}\), where \( v_{\text{rel}} = v_x - V_x \) is the velocity of the block relative to the plate. Initially the plate is at rest and the block, having just been hit with a hammer, has velocity \( v_{0x} = v_0 \). Find an expression for the time \( t_{1/2} \) at which the relative velocity \( v_{\text{rel}} \) has been reduced to half its initial value.