Question

A car with a total mass of 1800kg (including passengers) is driving down a washboard road with bumps spaced 4.4m apart. The ride is roughest

Answer 1

The force that the spring exerts is k*x, where x is the distance of compression.

If the spring is compressed an infinitesimal distance dx, the energy required to do this is:

dE = F*d = (k*x)*dx

If the spring is compressed a finite distance d, the total energy required would be given by the integral of this equation from x=0 to x=d:

E = integral of {k*xdx) from x=0 to x=d

= 1/2*k*x^2 (evaluated from x = 0 to x=d)

= 1/2*k*d^2

If the car is stopped completely, then all of its kinetic energy is captured by the spring, so

1/2*m*v^2 = 1/2*k*d^2, or

m*v^2 = k*d^2, or

k=m*v^2/d^2

= (1800kg)*((5.5)*(5.5)/(4.4)^2

=54450/19.36

= 2812.5 N/m