# Help with this problem about car speed

#### jeff5656

Lets say you are in a car going 60 mph and ahead of you is a flat bed truck (that you can drive on top) that is going 50 mph. There is someone standing along the road that can measure your speed (like with a laser gun, say.)
Now, as soon as you drive up on the truck, how fast are you going relative to the truck and to the ground? Your tires should immeditely "see" the truck as the ground and since the tires are going around at 60 mph at that instant, wouldn't it follow that you would go flying of th truck at 60 mph? If that is the case, wouldn't the person on the ground suddenly measure your speed at 120 mph? Obviously this can't happen, but nontheless, the tires ARE going 60 mph at the instant it touches the truck and the truck is stationary as far as those tires are concerned, because how do the tires "know" what the truck speed is relative to the road?
But lets say that as soon as you go on the truck your speed is 10 mph because the truck was going at 50 relative to the road. What, then, does your speedometer say at that very instant? Does it immediately decelerate to 10 mph? If this is the case, does the driver immediately experience a decelertion feeling from 60 to 10 mph?
If there is no deceleration, and the speedometer still says 60 but you are really only going 10 relative to the truck, then that implies that the car is still communicating with the ROAD frame of reference (because there is no deceleration). In other words, when you go from the road to the truck, you are instantly changing the frame of reference for the car.
SO: which is it? If the car's frame of reference is now the truck, you should decelerate to 10 mph and the speedometer shold reflect that OR f the frame of reference is still the road you you hould fly off the truck at 120 mph!

Can someone help to resolve this apparent pardox?

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#### brewnog

Gold Member
Firstly, forget about things like "instantly" and "suddenly". For this kind of situation, there are no "immediate" changes. This is the key to resolution of the apparent paradox. All changes will happen over a time period. I'm sure you're aware of this, but I thought I'd clarify.

Once the drive wheels of your car touch the ramp of the truck, several things will happen:

The speed at which the car is actually travelling, and the rate at which the drive wheels are trying to cover ground will not be equal. As you know, it takes time to accelerate or decelerate a tonne of car. In this situation, the motion of the car (as a result of its momentum) will essentially be "fighting" against the motion of the wheels, and as a result there will be a lot of skidding while the drive wheels of the car are forced to match the speed at which the car is moving with respect to the truck (which will be 10m/s, give or take a bit), and there would probably be some nasty crunching of the car's transmission, while it struggles to cope with the rapid deceleration which is being forced upon it by the rapid rotational deceleration of the car's drive wheels.

You must remember that tyres are capable of skidding on the ground, and that it takes a very large force to accelerate a tonne of car. The speedometer of the car will give you the reading for the instantaneous rate at which the drive wheels are turning. I would expect there to be a pretty swift deceleration from 60 to 10mph.

I realise that it's a pretty complex problem to convey in words, perhaps someone will give a more concise explanation than my rather hashed effort.

#### timberfella

Sorry, can't give an answer, but its always mystified me too!
Perhaps someone who worked on the making of "The italian Job" might be able to help?

#### Q_Goest

Homework Helper
Gold Member
Just to help elaborate on what Brewnog is saying, when an aircraft lands on a runway, the tires go from essentially zero miles an hour to however fast the airplane is going. In those few moments, you hear a loud chirp from the tires and a puff of smoke. The airplane doesn't suddenly stop on the runway because the tires are not spinning when it touches down, it takes a hell of a lot more than some non-rotating tires to stop an airliner! So what happens? The tires skid on the ground as they accelerate rapidly to the same speed the aircraft is moving. In so doing, the tires skid on the runway, and all the energy it takes to rotate them that fast creates the heat that results in a puff of smoke from the tires.

If you've ever flown in an aircraft, you also know that as much energy is needed to smoke the tires, the amount you are slowed down by those tires is small and negligable. The tires are way too small to rapidly decelerate the aircraft. You need brakes and thrust reversers for that.

Similarly, when the car's tires hit the ramp on the truck, you might hear a chirp and maybe even a bit of smoke (not likely) but you certainly won't see any significant change in the speed of either vehicle. The vehicles simply weigh too much to be affected by the relativly light weight tire.

If you want to get into it deeper than this, note that the translational energy of the vehicle is extremely large compared to the rotational energy of the tire. Conservation of energy would tell you that you can't change the vehicle's velocity with the small amount of energy available from the rotating wheel.

#### brewnog

Gold Member
Thanks Q, I knew I could rely on someone else to bring clarity to this thread!

#### matthewkeating

Also the flat bed truck would deccelerate ever so slightly. At last we can understand how "kit" from nightrider drove into the moving truck!

#### mpmendez

So, the stationary observer, after a period of time, would eventually measure the speed of the car to be 60 + 50? This is assuming that he has supplied the same amount of pressure on the accelerator pedal to accelerate to 60 mph while on the flatbed. However, initially, the stationary observer would see the car match and overcome the flatbed truck velocity as it negotiates the ramp. Does this make sense?

#### Q_Goest

Homework Helper
Gold Member
mpmendez, Just to make sure I understand what you're asking. If the flat bed truck had an exceeding long bed (ie: on the order of a mile or so), and if the car had topped out at 60 before hitting this ramp, and if the car's accelerator was fixed in that position, then it would eventually get to 60 mph again with respect to the flatbed truck with one big exception. In fact, the largest force on the vehicle is air drag, so the air drag would be much higher. If it were to go 60 mph on the back of the truck, and if the truck were going 50 mph with respect to the ground, the car would have to be going 110 mph and it would 'feel' the wind at that velocity. That's a whole lot more wind resistance than 60 mph. So in truth, no it would never get up to that speed. Wind resistance would prevent it from doing so, but otherwise the logic you provided is correct.

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