Consider with a car of mass 1000 kg moving with a speed 18.0 km/h on a road and colliding `6.25 xx 10^(3) Nm^(-1)`. Taking the coeffici-ent of friction, `mu` to be 0.5 what is the maximum compression of the spring ?

Explanation

In presence of friction, both the spring force and the frictional force act so as to oppose the compression of the spring. We invoke the work - energy theorem, rather than the conservation of mechanical energy.
The change in kinetic energy is
`DeltaK=K_(f)=0-(1)/(2)m u^(2)`
The work done by the net force is
`W=-(1)/(2)kx_(m)^(2)-mumgx_(m)`
Equating we have `(1)/(2)mu^(2)=(1)/(2)kx_(m)^(2)+mumgx_(m)`
Now `mu mg=0.5xx10^(3)xx10=5xx10^(3)N`
`("taking g" = 10.0s^(2))`. After rearranging the above equation we obtain the following quadratic equation in the unknown `x_(m)`.
`kx_(m)^(2)+2mu mgx+_(m)-m u^(2)=0`
`x_(m)=(-mu mg+[mu^(2)m^(2)g^(2)+mku^(2)]^(1//2))/(k)`
Where we take positive square root since `x_(m)` is positive. Putting in numerical values we obtainthe following quadratic equation in the unknown `x_(m) = 1.35m.`