Consider a simulation with a car of mass 1000kg moving with a speed 18.0 km/h on a road and colliding with a horizontally mounted spring of spring constant `6.25 xx 10^(3)N m^(-1)`. Taking the coefficient of friction, `mu` to be 0.5 What si 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 teh net force is
`W= -(1)/(2) kx_(m)^(2)- mu mgx_(m)`
Equating we have `(1)/(2) m u^(2) = (1)/(2) kx_(m)^(2)- mu mgx_(m)`
Now `mu mg = 0.5 xx 10^(3) xx 10= 5 xx 10^(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 mgx_(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`