Calculating Time for Rectangular Block to Stop on Rough Ground

[gaobo9109]

Homework Statement

A rectangular block of length L is traveling on a smooth surface at velocity u. When it encounters rough ground with coefficient of friction mu, it decelerates to a stop. What is the time taken for the rectangular block to stop?

Homework Equations

F= (mu)mg

The Attempt at a Solution

Deceleration of the block is given by a = (mu)gx/L if only part of the block is on the rough ground. If the entire block is on the rough ground, the maximum deceleration is given by a = (mu)g. This means i have to split the question into two cases.
1. time taken when x is less than or equal to L
2. time taken when x is greater than L

But I am stuck here. I don't know how to relate x to t and u.

 

[zorro]

Friction force is independent of area of contact. So I think there is no need for taking a part of the block.

deceleration of the block is (mu)g
apply kinematics equation
v^2 - u^2 =2as
v=0 s=L
find a relation between u and s from here
use another equation - s=ut +0.5at^2
find out time from this equation. ( here discriminant will become 0 from previous equation.)