Velocity of a trolley on an inclined plane with a twist

[sriram_15_93]

A friend asked me how to solve this problem:

There is a trolley of mass M$$_{o}$$kg on a fixed inclined plane, whose angle of inclination is $$\theta$$. The block is at a height Hmetres, at the top of the inclined plane. Coefficient of friction is $$\mu$$. It is also known that it is raining, and the rain drops are falling with a velocity ums$$^{-1}$$ at the rate of $$\eta$$ kgs$$^{-1}$$. The rain is also falling, making an angle $$\alpha$$ with the ground. The trolley is released from rest at t=0. Find the velocity v of the trolley when it reaches the bottom.

I solved this question and ended up with a big differential equation in terms of dv and dt. However, part of the equation contains a dx (displacement along the plane) with t in the denominator that I am unable to get rid of. The only way I can do so is by means of a relation between displacement and time. Could someone help me with this part?

The problems I'm facing are:
(i) acceleration is varying
(ii) velocity is varying
(iii) I'm unable to get a constraint independent of velocity of the trolley.

I've attached a .bmp of the figure. Hope it helps.

 

[jbsteele]

To solve the problem, you will need to break it down into smaller parts. Firstly, you will need to determine the forces acting on the trolley. This will include the force of gravity, the force of friction and the force due to the rain droplets. You will then need to use Newton's second law of motion to calculate the acceleration of the trolley. This will be a function of the velocity of the trolley and the forces acting on it. Once you have determined the acceleration, you can then use the equation for motion with constant acceleration to calculate the velocity of the trolley at any given time. This equation is:v = u + atWhere u is the initial velocity, a is the acceleration and t is the time.Finally, you can use the equation for displacement with constant acceleration to calculate the displacement of the trolley at any given time. This equation is:x = ut + 0.5at^2Where u is the initial velocity, a is the acceleration and t is the time.By combining these equations and solving for the velocity and displacement of the trolley at the bottom of the inclined plane, you will be able to determine the velocity of the trolley when it reaches the bottom.