Final answer:
To find the speed of the cube just after it leaves the incline, calculate the initial gravitational potential energy, subtract the work done by friction on the incline, and equate it to the kinetic energy at the bottom. Solve for the speed using the energy conservation equation.
Step-by-step explanation:
To determine the speed of the cube just after it leaves the inclined plane, we can apply the principles of energy conservation and account for the work done by friction. Initially, the cube has gravitational potential energy (GPE) due to its height, which gets partly converted to kinetic energy (KE) and partly lost to friction as it moves down the incline. The work done by friction is calculated using the force of kinetic friction and the distance traveled on the incline.
The GPE at the top is given by mgh, where g is the acceleration due to gravity, and h is the height of the incline. The force of friction (f) is μp ⋅ mg cos( θ ), and the work done by friction (W) over the distance of the incline is f ⋅ d, where d is the length of the incline, calculated as h / sin( θ ). So the energy just after leaving the incline is, total energy = initial GPE - work done by friction. Equating this to the KE at the bottom of the incline (1/2)mv^2 and solving for v gives us the speed of the cube just after it leaves the inclined plane.
When calculating these values, ensure that all units are consistent and that you are considering the correct trigonometric functions for the angles involved.