When driving over a washboard speed bumps, certain speeds make the bumps less "bumpy." Suppose your car has mass of 1000 kg and the critical damping constant of the shock absorbers is 12000 kg/s, what is the "ideal speed" to drive across the bumps assuming that the bumps are 0.3 m apart? Hint: Find the decay time for the critically damped shock absorbers.
A) 1.2 m/s
B) 2.4 m/s
C) 3.6 m/s
D) 4.8 m/s
E) 6 m/s

Question

When driving over a washboard speed bumps, certain speeds make the bumps less "bumpy." Suppose your car has mass of 1000 kg and the critical damping constant of the shock absorbers is 12000 kg/s, what is the "ideal speed" to drive across the bumps assuming that the bumps are 0.3 m apart? Hint: Find the decay time for the critically damped shock absorbers. 
A) 1.2 m/s 
B) 2.4 m/s 
C) 3.6 m/s 
D) 4.8 m/s 
E) 6 m/s

Quizplus · Accepted Answer

The decay time for critically damped shock absorbers is given by:

t = 1/ (ζωn)

where ζ is the damping ratio, ωn is the natural frequency (ωn = √(k/m)), k is the spring constant and m is the mass of the car.

With critical damping, ζ = 1, so

t = 1/ (ωn) = 1/√(k/m)

The distance between the bumps is given as 0.3 m. To minimize the effect of the bumps, we want the car to pass over them in a time less than the decay time of the shock absorbers. So, if the car is traveling at a speed of v,

t = 0.3/v

The ideal speed would be the one that makes the time of passage equal to the decay time:

t = 1/ (ζωn) = 1/ωn = 1/√(k/m)

Thus,
0.3/v = 1/√(k/m)

v = 0.3 × √(k/m) = 0.3 × √(12000/1000) = 2.4 m/s

Therefore, the ideal speed to drive across the bumps is B) 2.4 m/s.

When Driving Over a Washboard Speed Bumps, Certain Speeds Make