A block of mass m travels in a horizontal circle against the wall...

The normal force exerted by the wall on the block provides the necessary centripetal force for circular motion:

$N = \dfrac{mv^2}{R}$

The frictional force acting tangentially opposite to the direction of motion is:

$f = \mu N = \dfrac{\mu mv^2}{R}$

The tangential acceleration of the block is given by Newton's second law:

$m a_t = -f \Rightarrow m \left(v\dfrac{dv}{ds}\right) = -\dfrac{\mu mv^2}{R}$

$v\dfrac{dv}{ds} = -\dfrac{\mu v^2}{R}$

Separating the variables to integrate:

$\dfrac{dv}{v} = -\dfrac{\mu}{R} ds$

Integrating both sides for one complete revolution, where the distance covered is $s = 2\pi R$, and the velocity changes from $v_0$ to $v$:

$\int_{v_0}^{v} \dfrac{dv}{v} = -\int_{0}^{2\pi R} \dfrac{\mu}{R} ds$

\left[\ln v\right]_{v_0}^{v} = -\dfrac{\mu}{R} [s]_{0}^{2\pi R}$<br><br>$\ln\left(\dfrac{v}{v_0}\right) = -\dfrac{\mu}{R} (2\pi R)$<br><br>$\ln\left(\dfrac{v}{v_0}\right) = -2\pi\mu$<br><br>$v = v_0 e^{-2\pi\mu}