Unless otherwise stated, in the following problems we assume that the gravitational force is constant with g = 9.81 m/sec2 in the MKS system and g = 32 ft/sec2 in the U.S. Customary System.An object of mass 60 kg starts from rest at the top of a 45º inclined plane. Assume that the coefficient of kinetic friction is 0.05 (see 18). If the force due to air resistance is proportional to the velocity of the object, say, -3y, find the equation of motion of the object. How long will it take the object to reach the bottom of the inclined plane if the incline is 10 m long?

Solution:Step-1:In this problem we need to find the equation of motion of the object and we have to find how long will it take the object to reach the bottom of the inclined plane if the incline is 10m long.step-2:Given: The mass of an object (m)= 60kg . Inclination .Gravitational force is constant with and b = 3.Assume that , the coefficient of kinetic friction We know that, Step-3:Therefore, the force of gravity in the direction is Normal force Now,we have to find the value of .Clearly, is a differential equation .Step-4:Now , find the general solution of the equation.We know that,is the integrating factor.Multiply the above equation both sides with . Integrating both sides we get:, since Therefore,step-5:We know that at initial point v = t= 0 , then becomes., since .Therefore, C = (-131.78)Hence,We know that, Integrating on both sides we get:, since . step-6:We know that at initial point x = t= 0.So, Therefore, By using method of newton’s law:Consider, differentiate both sides with respect to t we get. Step-7:Therefore, If , then = 2- (0.2187) = 1.7814Similarly, =1.7814-0.0117 = 1.7697Therefore, t at x= 10 is 1.7697 seconds