How Fast Would NASCAR Cars Go at Daytona without Restrictor Plates?

Doug Yates was  guest on Dave Moody’s SiriusXM Speedway last week. He brought up a conversion you hear a lot in the week before Daytona and Talladega.  Every 25 horsepower in the engine translates to about a 1 second decrease in lap times. Dave did the math: Removing the plates would increase the engine by 450 horsepower. Four hundred and fifty more horsepower equates to 18 seconds off the lap time, assuming all other things equal.  That last part was a very important qualification. It will come back to haunt us in a moment.

David Gilliland got the pole at Daytona with a lap speed of 45.153 seconds, translating to a speed of 199.322 miles per hour. Using the above argument, his lap time would decrease to 27.153 seconds.  That translates to a speed of 331.456 mph.In 2004, Rusty Wallace ran 228 mph at Talladega in an unrestricted engine. That’s almost 100 mph slower than our theoretical max speed. Let’s ignore the concerns that NASCAR racecars tend to get aerodynamically unstable if they turn around at high speeds and think only about straight line speed.

So let’s look at what limits how fast a car can go.  We’re considering two major forces: the force the engine makes, which propels the car forward, and the drag force, which pushes the car backward.

 

It’s exactly like a tug of war. Which ever is pulling harder, that’s the direct the car is going to go. (That’s because force is, as your physical teacher no doubt repeated over and over, a vector.At most tracks, if you want to pass someone, you step on the gas, the engine produces more force and you accelerate.  Daytona and Talladega are unique in that engine power is limited to about 450 hp. The pedals on the floor the whole way around. You’re perennially in the last situation, in which the car moves at a constant (a terminal) speed, which is the fastest speed you can get. The engine is doing all it can.

So here’s the catch – the reason why the phrase “all other things equal” causes problems.  All other things are not equal. Specifically, drag. Drag is simply the force of the air molecules pushing on the car, but that force increases quadratically, just like downforce. If the car goes twice as fast, you get four time (two squared) the drag.  It’s not fair: you have to work four times harder to get twice as fast. But that’s physics for you.

The argument above relied on the assumption that the drag remained constant – and it definitely doesn’t. It gets worse, because power depends on velocity cubed. So to go twice as fast, you have to overcome four times as much drag and you need eight times (2x2x2) the power.

You can estimate the terminal velocity of a racecar using some simple physics. The terminal velocity is the ratio of the power (P) to the drag (D):

The maximum drag you can have is proportional to the terminal velocity squared:

Which means that the terminal velocity ends up depending on the cube root of the power!

If the power of an engine doubles, the terminal velocity only increases by the cube root of two, which is 1.26. If we take 200 mph as a nice round terminal velocity for a restricted engine, removing the plates and doubling the engine power to 900 hp would only increase the terminal velocity for the unrestricted engine to 252 mph.  That’s pretty surprising – you double the engine power and you only get 50 mph more.  Such is the power of cube roots.

If nature were linear, it would be a whole lot less interesting.

Many thanks to my friends Josh Browne and Andy Randolph, both excellent engineers and always willing to let me bounce ideas off them and verify that I’m not crazy.  Not, at least, when it comes to physics.