# How to find time, acceleration under this situation

I am trying to solve the following.

An object of mass $m$ slides on a horizontal surface with initial velocity $v_0$. If the kinetic friction between the object and mass is given by $\mu$, find the following.
a) the time $t$ necessary to stop the object from moving.
b) the distance $d$ that the object travels before it comes to stop.
c) the acceleration $a$ of the object during this process.

The reason why I am asking this is because I am actually given $m,v_0$ and $\mu$ in order to solve for these values, but I am not sure how to find any of them because they seem to be all connected with the other unknown values.

What I know so far is the fact that the only force working on the object is the force of friction $F_{\mu}=-mg\mu$.

I know that $\Delta v = at$, but we have two unknowns.

${v_f^2-v_0^2\over{2a}}=d$ also has two unknowns.

What am I missing here ?

You should realize that the first equation you write gives you the value of $a$.

In these kinds of problems, you are always given some force, and you are expected to apply Newton's laws to the problem. Therefore

$F_{\mu} = -mg\mu \underbrace{=}_{\text{$2^{nd}$Law}} ma \quad \quad \to \quad \quad a = -g\mu$

Now you know the acceleration, you can find how the speed of the object evolves

$v(t) = v_0 + at = v_0 -g\mu t$

Therefore, the time needed to stop is the time after which a null speed is reached

$v(t_{stop}) := 0 = v_0 + at_{stop} \quad \to \quad t_{stop} = \frac{v_0}{g\mu}$

Now you have all the ingredients to find the distance travelled from the beginning until the object stops. Indeed

$x(t) = x_0 + v_0t + \frac{1}{2}at^2$

They're asking you the distance required to stop, which is $x(t_{stop}) - x_0$.

$d := x(t_{stop})-x_0 = v_0\cdot \frac{v_0}{g\mu} - \frac{1}{2}g\mu \frac{v_0^2}{(g\mu)^2} = \frac{v_0^2}{2g\mu}$

I have shown you all the reasoning that needs to be done to answer these kind of questions. Refrain from using formulas to calculate for example the stopping distance. Always make sure you understand when they come from and make sure you can rederive them.

Remember the law $F=ma$; you already know the force from friction $F_{\mu}= -mg\mu$. Hence you can get $a=-g \mu$, and one of your unknowns is gone.