Car on bumpy road

Bumpy road

A car with 4 passengers travels on a rough road with “bumps” at a distance $d$ from each  other (a bumpy road). Because of these bumps the car begins to bounce on its suspensions, with maximum amplitude when the speed is $v$. Without the 4 passengers the car increases its height from the road to $h$. Please find $h$.

bumpy road

Continue reading on physics of roads with bumps….

The passengers of the card and the car itself situated on the bumpy road make a harmonic oscillator. When the car is in motion the period of the oscillation is equal to the time to travel between two bumps of the road.
$T = d/v$
For a harmonic oscillator
$T = 2*pi*\sqrt {(4m+M)/k}$
$k =4*\pi^2* (4m+M)/T^2 =4*\pi^2* (v/d)^2 *(4m+M)$
In the fully loaded situation the suspensions are pressed with a distance $x1$ given by
$(M+4m)*g = k*x1$
When the car is unloaded the suspensions are pressed with a distance $x2$ given by
$M*g= k*x2$
As a result the height the car rises when is unloaded is
$h = x2-x1 = (4mg) / k=(4mg)/(4*\pi^2) * (d/v)^2* 1/(4m+M)$

$h = [d/(\pi*v)]^2 * [mg/(M+4m)]$