Wave speed of the collision process in a line of cars

[snickersnee]

Homework Statement

I need to write an equation for the at rest at a stoplight. Each car bumps into the one in front of it until the first car in line gets bumped.

Homework Equations

I found these equations for wave speed:
V=√(T/(m/L)), where T is tension, m is mass, L is length
and of course V=λf

The Attempt at a Solution

But both those equations have a time factor, and I wasn't given any time information. All I know is that wave speed needs to approach infinity as x → 0, and wave speed approaches v_o as x→∞
Any hints would be appreciated.

 

[mfb]

I found these equations for wave speed:
V=√(T/(m/L)), where T is tension, m is mass, L is length

You don't have a string here.

I think you are supposed to assume that all collisions are either perfectly inelastic or elastic. The latter is easier to study: what is the speed of the first bumped car after the collision? How long does it take to make the next collision?

 

[snickersnee]

Yes we're assuming elastic collisions. The first bumped car is traveling at v₀ because the car behind it transferred all its energy. The time until the next collision is

. I was thinking the total distance would be (4L+3x) and total time would be 3v₀/(L+x) but then (total distance)/(total time) gives (4L^2+7xL+3x^2)/3v₀ but that doesn't satisfy the limits required.

 

[mfb]

Why should the time change for longer cars (larger L) if the distance between the cars does not change?

Also, your fraction does not have units of time, it has units of inverse time.

 

[snickersnee]

Why should the time change for longer cars (larger L) if the distance between the cars does not change?

Also, your fraction does not have units of time, it has units of inverse time.

Thanks for your help. I think the time should change for longer cars because the disturbance takes longer to pass through that car, assuming the velocity of the disturbance is constant. And you're right, I should flip the fraction so it becomes

 

[mfb]

If the shock would travel at a speed of v (the same speed as the final car speed), your cars would get completely compressed to a single "disk" in the process - certainly not realistic.
Assume that the cars are incompressible (they cannot get deformed permanently in elastic collisions anyway) - as soon as the back of a car moves, the front moves as well.