# How long does it take a block to slide up and down a plane with friction?

So I am studying for my exam on monday for analytical mechanics. I am kind of stuck on a question, so wondering if someone can give me a pointer. I am actually looking for some pointers and some techniques (hints and tips) to get me through this. the topic is just simple velocity as a function of time, velocity as a function of displacement, quadratic air resistance, oscillations.

So here is the question:

A block of wood is projected up an inclined plane with initial speed $v_0$. If the inclination of the plane is $30^\circ$ and the coefficient of sliding friction is $\mu_k = 0.1$, find the total time for the block to return to the point of projection.

Heres my work: $$F_{\text{net}} = -F_f = -\mu_k \cdot m \cdot g \cdot sin(30) = m \cdot \frac{dv_x}{dt}$$ so the m's cancel and solving the differential equation you get the following: $$\int_{v_0}^v dv = \int_0^t -\mu_k \cdot g \cdot sin(30) dt$$ $$v-v_0 = -\mu_k \cdot sin(30) \cdot g\cdot t$$ so $$v(t) = v_0 - 0.98\cdot sin(30) \cdot t$$

so if i set that to 0 i know the time when it reaches the top and starts to slide backwards. Now how would I know when it reaches the point of projection? Would it just be double the time?

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Gravity force (in the direction of sliding) is $mg\sin(30)$. Friction force is therefore $0.1mg\cos(30)$ and opposes velocity. From these you get the accelerations (different for going up and down). Now move the block up to its turning point using the going up acceleration (which is the one with the largest magnitude), and from its position, figure out the time to get back down using the coming down acceleration.