Question

A particle of mass m and velocity $v_1$ in positive y direction is projected on to a belt that is moving with uniform velocity $v_2$ in x-direction as shown in figure. Coefficient of friction between particle and belt is $\mu .$ Assuming that the particle first touches the belt at the origin of fixed x-y co-ordinate system and remains on the belt, find the co-ordinates (x, y) of the point where sliding stops.

Answer 1

The problem can be divide into two parts of application of kinematic equation, $i.e.$ along x-axis and y-axis.
Along x-axis,
Initial velocity($u_x$) = $0m/s$
Final velocity, when it stops sliding along x-axis, ($v_x$) = $v_2$
Let x coordinate after sliding stops be $x$.
Since $\mu$ is the coefficient of friction, force of friction between belt and mass($m$) = $\mu mg$
Acceleration due to friction = $\mu g$
Using third kinematic equation, $v_2^2=2\mu gx$ and therefore $[x=\frac{v_2^2}{2\mu g}]$.
Along y-axis
Initial velocity($u_y$) = $v_1$
Final velocity, when the sliding stops($v_y$) = $0m/s$
Acceleration due to friction = $-\mu g$.
The y coordinate when sliding stops be $y$.
Using the third kinematic equation, $0=v_1^2-2\mu gy$ and therefore $[y=\frac{v_1^2}{2\mu g}]$