Final answer:
To find the velocity of the box when it reaches the bottom of the ramp, you can use the concept of conservation of mechanical energy. The work done by the friction force can be equated to the difference in potential and kinetic energy to calculate the velocity.
Step-by-step explanation:
To find the velocity of the box when it reaches the bottom of the ramp, we can use the concept of conservation of mechanical energy. The work done by the friction force is equal to the initial potential energy of the box at the top of the ramp minus the final kinetic energy of the box at the bottom of the ramp.
The work done by the friction force is given by the equation W = Fd, where F is the frictional force and d is the distance. So, W = (6.0 N)(6.0 m) = 36.0 Joules.
Using the equation for gravitational potential energy, PE = mgh, we can determine the initial potential energy of the box at the top of the ramp. PE = (40.0 N)(6.0 m)(sin 30.0°) = 120 Joules.
Finally, we can solve for the final kinetic energy using the equation KE = 0.5mv^2, where v is the velocity of the box at the bottom of the ramp. KE = 0.5(40.0 N)v^2. We can equate the work done by the friction force to the difference in potential and kinetic energy, resulting in the equation:
36.0 J = 120 J - 0.5(40.0 N)v^2
Simplifying the equation will give us the value of v, which is the velocity of the box when it reaches the bottom of the ramp.