Final answer:
To find the initial speed of the box, we can use the work-energy theorem. The work done by the spring and the work done by friction are both involved in the calculation. By applying the given values to the equation and solving for the initial speed, we can determine its value.
Step-by-step explanation:
To find the initial speed of the box, we can use the work-energy theorem. The work done on the box is equal to the change in its kinetic energy. The work done by the spring is equal to the potential energy stored in the spring when it was compressed. The work done by friction is equal to the friction force multiplied by the distance of sliding.
Let's denote the initial speed of the box as vi, the final speed as vf, the spring constant as k, the distance the spring is compressed as x, and the coefficient of kinetic friction as μk.
According to the work-energy theorem, the work done on the box is equal to the change in its kinetic energy:
Work done by spring + Work done by friction = Change in kinetic energy
Using the equation for the work done by the spring:
(1/2)kx² + μkmgx = (1/2)mvf² - (1/2)mvi²
Substituting the given values and solving for vi:
vi = sqrt((kx² - 2μkmgx)/(m + (k/m)))