Can You Solve These Challenging Friction Problems?

[Whitebread]

I don't know how to solve these questions and I was wondering if anyone here could help me

First question
The coefficient of static friction between hard rubber and normal street pavement is about .8. On how steep a hill (maximum angle) can you leave a car parked?

Last one
A motorcyclist is coasting with the engine off at a steady speed of 20m/s but enters a sandy stretch where the coefficient of friction is .80. Will the cyclist emerge from the sandy stretch without having to start the engine if the sand lasts for 15m? If so, what will be the speed upon emerging?

Could the person not give me the answer, but show me how it's done? I would be VERY greatful (preferably with a sum of forces in X and Y directions as I find that easier to understand, but I feel inconsiderate at this point so it's not necessary). Thank you.

 

[airbuzz]

First: if you use breaks then you have that the maximum static force is given by
$$\mu N\cos\vartheta$$
and the parallel to street component of the car weight N is
$$N sen\vartheta$$
You start to move when the second is bigger than the first, so the maximum angle is given by
$$Nsen\vartheta=\mu N\cos\vartheta\Rightarrow\tan\vartheta=\mu\Rightarrow\vartheta=\arctan\mu$$

 

[airbuzz]

Second: you must use the Newton´s law. Acceleration equals force so that if $$x$$ is space then speed and acceleration are $$\dot{x}$$ and $$\ddot{x}$$ where the dots mean derivation in time. If $$\mu$$ is the friction coefficient the Newton´s law gives
$$m\ddot{x}=-\mu mg\rightarrow\ddot{x}=-\mu g$$
where g=9,81
You must integrate two times in dt with the conditions
$$\dot{x}_0=V_0$$
$$x_0=0$$
So you find
$$\dot{x}=V_0-\mu gt$$
Then you stop after a time
$$t=\frac{V_0}{\mu g}$$
and
$$x=V_0t-1/2\mu gt^2$$
in which you must substitute the t found. So you find if x is longer than the sand or not.
Bye

 

[airbuzz]

Then, to find the speed upon emerging you must substitute the sand length [15 m] in the x formula, and find the time you need to emerge from the sand. When you find this you substitute this t in the speed formula ($$\dot{x}$$) and find the emerging speed.
And you have finished all the problems.

 

[Whitebread]

Thank you very much!