Consider a simulation with a car of mass 1000 kg moving with a speed 18.0 km/h on a road and colliding with a horizontally mounted spring of spring constant `6.25xx10^(3)Nm^(-1)`. Taking the coefficient 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 chenge in kinetic energy is
`Delta K = K_(f)-K_(i)=0-(1)/(2) m u^(2)`
The work done by the net force is
`W=-(1)/(2)kx_(m)^(2)-mu mg x_(m)`
Equating we having `(1)/(2)m u^(2)=(1)/(2)kx_(m)^(2)+mu mg x_(m)`
Now `mu mg = 0.5xx10^(3)xx10=5xx0^(3)N`
(taking `g = 10.0 ms^(-2)`). After rearranging the above equation we obtain the following quadratic equation in the unknown `x_(m)`.
`kx_(m)^(2)+2mu m g x_(m)-m u^(2)=0`
`x_(m)=(-mu mg +[mu^(2)m^(2)g^(2)+mku^(2)]^(1/2))/(k)`
where we take the positive square root since `x_(m)` is positive. Putting in numerical values we obtainthe following quadratic equation in the unknown `x_(m)=1.35m`.