## Fundamentals of Physics Extended (10th Edition)

$0.21m$
Lets assume that $H_{0}$ is initial height of the box relative to the ground. The total energy of the spring and box system initially will be: $E_{tot}=mgH_{0}..............(1)$ Since there is no speed of the box initially and the spring is relaxed when the box starts to move, its potential energy will be converted to the box’s kinetic energy and spring’s potential energy. Lets assume that the box will descend a distance of $L$ on the incline (so the spring will stretch by distance $L$) and lets write the total energy equation again: $E_{tot}=E_{spring}+E_{box}=\dfrac {kL^{2}}{2}+\dfrac {mv^{2}_{1}}{2}+mgH_{1}............(2)$ $H_{1}$ is the height of the box relative to the ground after descending $L$ distance. From the incline, we get: $\Delta H=H_{0}-H_{1}=L\sin \theta \left...........( 3\right)$ From (1) and (2), we get: $mg\left( H_{0}-H_{1}\right) -\dfrac {kL^{2}}{2}=\dfrac {mv^{2}_{1}}{2}\Rightarrow v_{1}=\sqrt {2g\left( Ho-H_{1}\right) -\dfrac {k}{m}L^{2}}\left....( 4\right)$ From (3) and (4) we get: $v_{1}=\sqrt {L\left( 2g\sin \theta -\dfrac {kL}{m}\right) }(5)$ So if the block momentarily stops then $v_{1}=0\left.....( 6\right)$ From (5) and (6), we get $L=\dfrac {2mg\sin \theta }{k}\approx 0.21m$