Relative speed

Date: Wed, 30 Nov 1994 12:35:30 -0500 (EST) 
From: "David G. Kolodny"
Subject: Relative Speed

This is a mathematic problem that I've mentally played with for some time
but do not, as yet, have a definitive answer. 

Two vehicles are travelling at 60 mph.  The first vehicle is a flatbed
truck.  The back of the truck has a drive-on ramp that is lowered down to
the roadway.  The flatbed trailer portion of the truck is 50 feet long. 

The second vehicle is a rear wheel drive sedan and is only 3 feet
directly behind the flatbed truck.  It is capable of coming to a complete
stop from 60 mph in 120 feet. 

The truck slows to 59 1/2 mph.  Since the car is gaining on the truck at a
rate of 1/2 mph, the front wheels of the car begin to slowly move up the
ramp. 

Question:

When the rear tires finally make direct contact with the ramp, will the
car which was moments ago traveling at 1/2 mph relative to the truck's
ramp, suddenly go from 1/2 mph ramp speed to 60 mph ramp speed and
thus crash into the back of the truck's cab section? 

Thanks!

David G. Kolodny

Date: 7 Dec 1994 01:07:50 GMT
From: Dr. Math
Organization: Swarthmore College
Subject: Re: Relative Speed

Hello there!

Your question brings to mind the old television show from my youth, 
Knight Rider, in which Kitt the supercar did exactly what you described 
on a regular basis.  They showed actual footage of the car driving along at
high speeds behind a semi-truck, and then rolling smoothly up the ramp the
semi extended.  It looked almost effortless.  So my instinct is to tell
you that yes, it's possible, because I've seen it done.

However, I decided when I was pretty young that this was a physical
impossibility, that they had either used some kind of trick photography,
or some fancy kind of car that could switch front- and rear- wheel drive. 
In any case, I think that your question is well-posed.  If the car
requires 120 feet to stop, then it will crash into the back of the truck
shortly after its rear tires hit the ramp.

But I rethought the problem, and now I can think of a couple of things
that might save the car and the driver.

1)  It seems to me that the main reason the car needs so much room to stop
is that it needs to kill its momentum.  It needs to go from 60 mph to 0
mph.  But when it drives up the ramp and onto the truck, it's only going
from 60 mph to 59.5 mph.  So I think it's going to take WAY LESS distance
to stop that car, because the only significant momentum that you need to
halt is the rotational momentum of the tires.  They do have to go from 60
mph to almost 0 mph, and they have to do it fast.  But the tires and axels
and stuff aren't nearly as massive as the whole car, so it shouldn't take
that much to stop them.

Now we see that this situation is essentially the same as jacking up a
car, revving the wheels, and seeing how fast they can stop.  Just don't
try to use your hands to stop them.

2)  When the tires first hit the ramp, there will be a short time when the
tires are touching both the ramp and the street.  This will give it a
little time in which the wheels can slip on the ramp and slow down,
without shooting the car through the back of the truck.

So I think it can stop pretty easily, particularly if the driver of the
car puts in the clutch, and doesn't make the engine go to 0 rpm in a quick
way.  That would only add more momentum of which to get rid.

I hope this helps your thinking some.  I wish there were some kind of "Dr.
Physics" around here to check my work.