An object of mass 5 kg is initially at rest on the surface. The surface has coefficient kinetic friction `mu_s` = 0.6. What initial velocity must be given to the object so that it travels 10 m before coming to rest?

Explanation

When the object moves on the surface it will experience three forces.
(a) Downward gravitational force (mg)
Upward normal force (N)
(c) Frictional force opposite to the motion of the object.
Since there is no motion along the vertical direction, magnitude of normal force is equivalent to the magnitude of gravitational force.
N = mg
Applying Newton.s second law along the x direction `mveca = - m_k mg hati`
The acceleration is `a = -mu_s mghati`
Note that the acceleration is along the x direction since the frictional force acts along the negative x direction.
` a = - mu_g k`
Note that the acceleration is uniform during the entire motion. We can use Newton.s kinematic equation to find the final velocity. Along the x direction `v_2 = u_2 + 2as`
Here v = final velocity and u = initial velocity to be given to travel a distance s.
In this problem s = 10 m
Since the particle comes to rest, the final velocity v = 0
`0 = u^2 - 2mu_k gs rArr u = sqrt(2mu_k gs)`
`u = sqrt(2 xx 0.6 xx 9.8 xx 10) = 10.8 ms^(-1)`