Final answer:
The final speed of a box sliding across a rough section can be found using the work-energy principle, considering the initial speed, the length of the rough patch, and the coefficient of kinetic friction.
Step-by-step explanation:
The problem involves calculating the final speed of a box sliding across a rough section of the floor using energy considerations. We know the initial speed (vi = 5.30 m/s), the length of the rough section (d = 3.60 m), and the coefficient of kinetic friction (μk = 0.100). Since there is no acceleration involved and we are neglecting air resistance, we can use the work-energy principle to solve this problem.
The work done by the frictional force (Wf) which is opposite to the direction of displacement is given by Wf = - * d, where is the kinetic friction force and can be calculated using = μk * m * g. Since we don't know the mass (m) of the box, we can just consider the work done per unit mass. The work-energy principle states that the total work done by all the forces acting on an object is equal to the change in kinetic energy (KE) of the object.
Thus, we have:
-(μk * m * g) * d = 0.5 * m * vf2 - 0.5 * m * vi2
The mass (m) can be canceled out from both sides of the equation, and we can solve for the final speed using:
0.5 * vf2 = 0.5 * vi2 - μk * g * d
Solving for vf gives us the final speed after crossing the rough section of the floor.