How Can I Solve This Dynamics Problem Involving Friction and Inclined Planes?

[JakePearson]

hey guys, I'm in the middle for revising for a dynamics paper in January. I'm in the middle of a 1st year BSc Physics degree. Looking through some past papers, there are questions that i can do, and there are questions that i am unable to do. These are usually question where you have to derive. I am going to write down one of the questions that are causing me problems and i was wondering if you could help me on how to answer it.

Q. An object moving along a horizontal surface reaches the foot of a plane inclined at an angle $$\vartheta$$ to the horizontal with a speed u. If the coefficient of friction between the object and the plane is $$\mu$$ show that the object will come to rest a distance d up the plane where

d = u² / 2g (sin$$\vartheta$$) + $$\mu$$cos$$\vartheta$$​

I am a diagnosed dyslexic, not using it as an excuse, i will always try my best, but this is not sinking in, can you please help.

 

[dE_logics]

We have to calculate the net retardation.

If you have the mass of the body and it's initial velocity, you can calculate it's rate of retardation (which will be a function of the friction and gravity) and finally the time/distance at which v = 0.


Now...you got to manipulate the equation that comes to the proof.

 

[JakePearson]

i can appreciate you comment. But this is what i mean. This isn't sinking in. My lectures speak to me like this. Could you work through it with me. Cheers

 

[MaxwellsDemon]

Imagine I give you a similar problem but without an incline...just one dimensional linear motion. So in this problem you have an initial velocity u and a constant deceleration -a...derive an equation for me predicting the stopping distance in terms of just those two variables. Your problem uses the same basic derivation, but now you need to figure out what -a is in your problem.

 

[Doc Al]

Q. An object moving along a horizontal surface reaches the foot of a plane inclined at an angle $$\vartheta$$ to the horizontal with a speed u. If the coefficient of friction between the object and the plane is $$\mu$$ show that the object will come to rest a distance d up the plane where

d = u² / 2g (sin$$\vartheta$$) + $$\mu$$cos$$\vartheta$$​

Be careful with parentheses. The answer should be:
d = u²/2g(sinθ + μcosθ)

There are two ways to approach this problem:
(1) Using straight dynamics. Hint: What forces act on the object? What's is its acceleration as it goes up the ramp? Use kinematics.
(2) Using conservation of energy. Hint: How much energy is 'lost' due to friction?

Pick one of those approaches and give it a shot. Show your work and you'll get plenty of help.