Question

A car moves with a constant tangential acceleration $w_{\tau }=0.62m/s^2$ a horizontal surface circumscribing a circle of radius $R=40m$. The coefficient of sliding friction between the wheels of the car and the surface is $k=0.20$. The distance(in meters) will the car ride without sliding if at the initial moment of time its velocity is equal to zero is 10x. The value of x is:

Answer 1

As initial velocity is zero thus
$v^2=2w_{t}s$ (1)
As $w_t>0$ the speed of the car increases with time or distance. Till the moment, sliding starts, the static friction provides the required centripetal acceleration to the car.
Thus $fr=mw$, but $fr\le kmg$
So, $w^2\le k^2{g}^2$ or, $w_t^2+\frac{v^2}{R}\le k^2{g}^2$
or, $v^2\le (k^2{g}^2-w_t^2)R$
Hence $v_{max}=\sqrt{(k^2{g}^2-w_t^2)R}$
so, from equation (1), the sought distance $s=\frac{v_{max}^2}{2w_t}=\frac{1}{2}\sqrt{{\left(\frac{kg}{w_{\tau }}\right)}^2-1}=60m$.