You take a 3 kg metal block and slide it along the floor, where the coefficient of friction is only 0.4. You release the block with an initial velocity of $\langle 6, 0, 0\rangle m/s$. How long will it take for the block to come to a stop? How far does the block move?

Mass of metal block = 3 kg

The coefficient of friction between floor and block = 0.4

Initial velocity of block = $\langle 6, 0, 0\rangle m/s$

$\Delta \vec{p} = \vec{F}_{net} \Delta t$

$ x: \Delta p_x = -F_N\Delta t $

$ y: \Delta p_y = (F_N - mg)\Delta t = 0 $

Combining these two equations and writing $ p_x = mv_x $, we have

$ \Delta(mv_x) = -mg\Delta t $

$ \Delta(v_x) = - g\Delta t $

$ \Delta(t) = \dfrac{0 - v_{xi}}{-g} = \dfrac{v_{xi}}{g} $

$ \Delta(t) = \dfrac{6 m/s}{0.4 (9.8 N/kg)} = 1.53s $

Since the net force was constant, $v_{x,avg} = (v_{xi} + v_{xf})/2$, so

$ \Delta x/\Delta t = ((6 + 0)/2) m/s = 3m/s $

$ \Delta x = (3 m/s)(1.53 s) = 4.5m $