(a) A box of mass m slides down an inclined plane that makes an angle θ with the hor...

(a) A box of mass m slides down an inclined plane that makes an angle θ with the horizontal as shown in FIGURE 2.7.11. Find a differential equation for the velocity v(t) of the box at time t in each of the following three cases:

(i) No sliding friction and no air resistance

(ii) With sliding friction and no air resistance

(iii) With sliding friction and air resistance

In cases (ii) and (iii), use the fact that the force of friction opposing the motion of the box is μN, where μ is the coefficient of sliding friction and N is the normal component of the weight of the box. In case (iii) assume that air resistance is proportional to the instantaneous velocity.

(b) In part (a), suppose that the box weighs 96 pounds, that the angle of inclination of the plane is θ = 30°, that the coefficient of sliding friction is and that the additional retarding force due to air resistance is numerically equal to Solve the differential equation in each of the three cases, assuming that the box starts from rest from the highest point 50 ft above ground.