# Is a car's top speed the same on a wet road?

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1. Feb 2, 2016

### electrodruid

I'm a games programmer who is writing a simple car physics simulation, and I've had a disagreement with the designer about a car's behaviour - specifically, its top speed - in different conditions.

Imagine a car accelerating from a standing start to top speed along a perfectly straight, flat road. We know the car's mass, so we can calculate acceleration at any given moment using a = F/m ... To calculate the forces I'm assuming that the car is resisted by a drag force (proportional to the car's velocity squared), and rolling resistance (proportional to the car's velocity), and is accelerated forward by a tractive force. I have a function that returns a traction force for a given velocity. It's arrived at by slightly convoluted means and is meant to represent high torque in lower gears, decreasing as the car speeds up and the driver changes up through the gears, and it looks kind of like a graph of y = 1/x

So the optimum tractive force that can be applied at any given moment is proportional to the coefficient of friction between the tyres and the road, right? For good quality tyres on dry asphalt we can say that the coefficient is 1.0. Assuming we plug in sensible values, we get a decent-looking acceleration. Now we re-run the simulation when the road is wet and the friction coefficient drops to 0.7 ... I think that that will scale the usable traction force at any moment to 70% of what it could be in dry conditions (more than that would cause the car to wheelspin), which means not only a slower acceleration but also a slower top speed. The designer agrees that acceleration would be slower but thinks that given a long enough road the car could eventually reach the same top speed in the wet as it can in the dry.

Who is right? And if he is right, how can that be explained and simulated?

2. Feb 2, 2016

### Bystander

Neither. You are less in error than he; rolling resistance on wet road is higher than on dry. All other things equal, horsepower output, the higher drag slows the car.

3. Feb 2, 2016

### jbriggs444

The coefficient of static friction should act to limit the maximum tractive force that can be applied. Your tractive force should have a flat line maximum with a value that depends on the coefficient of static friction and vehicle mass.

4. Feb 2, 2016

### A.T.

It depends on whether the maximal speed on the dry road is determined by max. engine power or static friction coefficient. If it's the later, then obviously any reduction of the coefficient will reduce the maximal speed. But if it's the max. power, then you can reduce the friction coefficient by a certain amount, without any consequences for maximal speed, until you reach a level where friction coefficient becomes the factor limiting maximal speed. The same argument applies to maximal acceleration.

If you are writing a simulation anyway, you can plug in different parameters and see when they affect top speed.

5. Feb 2, 2016

### Staff: Mentor

My suspicion is that for most real cars, the static friction coefficient stops being a driving factor by the time you get out of first gear.

6. Feb 2, 2016

### A.T.

In the first gear it's a limiting factor for acceleration. But for most normal cars on a dry or slightly wet road its probably not the limiting factor for speed. If the road is flooded, that creates it's own drag limiting the speed. I can imagine static friction coefficient limiting speed on ice and compressed snow.

7. Feb 2, 2016

### electrodruid

Interesting. So to summarise, there's a threshold speed/gear above which the friction coefficient becomes effectively irrelevant and the car's top speed is instead limited by rolling resistance (or some additional drag force) instead? That raises some questions:

- How might you go about modelling that effect of the static friction? Presumably the point at which the limiting factor is drag rather than friction changes depending on the surface type (we're trying to model a bunch of different surfaces: wet/dry asphalt, dirt, mud, gravel, sand, snow, ice, etc), and presumably that has something to do with how much tractive force is available at a given gear/speed but unused because the friction coefficient limits how much force would be useful. But I'm having trouble visualising how that all fits together.

- Similarly, how might you model this drag/rolling resistance force? Is it like drag (i.e. proportional to velocity squared), or more like rolling resistance (i.e. proportional to velocity), or something else? And how might it differ for different surface conditions?

8. Feb 2, 2016

### jbriggs444

The limit on top speed is the point at which thrust is equal to drag. It is the speed at which the graphs of drag-as-a-function-of-speed and thrust-as-a-function-of-speed intersect.

This point may occur in a region where thrust is friction-limited (e.g. a car driving on ice) or in a region where thrust is engine-limited (e.g. a ordinary car on dry pavement).

9. Feb 2, 2016

### A.T.

For most cars on a road it's mostly aerodynamic drag vs. propulsive power.

It limits how much propulsive force is possible

The total drag is the sum of the individual drags.

10. Feb 2, 2016

### electrodruid

Understood. Perhaps I need to clarify how I'm currently doing things. So I've calculated the propulsive and resistive forces for a car on a dry drag strip - friction coefficient of 1.0. So for a wet drag strip I'm just looking up those same numbers I calculated earlier, but multiplying the traction force for a given velocity by 0.7 (wet road friction coefficient) to simulate that limiting factor. Leaving the drag and rolling resistance forces the same, relative to velocity, and just running the simulation. What I'm gathering from this conversation is that this is probably the wrong approach since limiting the traction force in this way can limit the top speed (i.e. the speed at which the scaled-down tractive force is equal to the sum of the unscaled-down drag) in quite a dramatic way.

Understood. TotalDrag = Aerodynamic Drag + Rolling Resistance + Wet Surface Drag. I wasn't previously aware that a wet surface (presumably also mud, ice, gravel, etc) can impose a new individual drag force, so I'm trying to work out what kind of shape that drag might take. Is it constant, linear, quadratic...?

11. Feb 2, 2016

### jbriggs444

How does the friction coefficient of 1.0 figure into your calculations? From everything you say, it appears that you are including it incorrectly.

12. Feb 2, 2016

### A.T.

The coefficient of static friction gives you the maximal static friction possible, not the actual static friction. The traction force needed for a given velocity is determined by total drag-

13. Feb 2, 2016

### electrodruid

For a given velocity v, my force calculation currently looks something like this:

F = (TractionForce(v) * frictionCoefficient) - ((DragCoefficient * v * v) + (RRCoefficient * v))

So with a friction coefficient of 1.0, i.e. a dry road, the available tractive force isn't scaled or limited in anyway. The car's acceleration is purely a result of the tractive and drag forces. For friction coefficients of < 1.0 the tractive force is limited, resulting in slower acceleration and a slower top speed.

14. Feb 2, 2016

### jbriggs444

As suspected, that calculation is incorrect. The effect of the coefficient of static friction is to impose a maximum force equal to vehicle mass times the acceleration of gravity times the coefficient of friction. That is better modelled as:

Thrust = min(TractionForce(v), m * g * frictionCoefficient)

Edit: min, darnit.

Last edited: Feb 2, 2016
15. Feb 2, 2016

### electrodruid

Ah! Perfect! Thank you!

Okay, so what about the other part - the idea that there's some kind of change or addition to the drag forces on wet or slippery surfaces, so the top speed of the car is perhaps still affected somewhat compared to dry conditions (but not affected directly by the friction)? What might that look like, as a function of velocity?

16. Feb 2, 2016

### Staff: Mentor

You need to build a model that relates engine power to tractive force. If you do a forum search, we've had several threads describing how to do it.

17. Feb 2, 2016

### jbriggs444

Note edit to post #14. It should have been min, not max.

18. Feb 2, 2016

### electrodruid

Hehe. I always make that mistake first time around, too :)

19. Feb 2, 2016

### electrodruid

My simulation doesn't really include engine power. For long, complicated, and not-terribly-relevant-to-my-question reasons, the tractive force is approximated from a sort of idealised mathematically (rather than physics-ally) constructed acceleration curve and the resistive forces. I don't really have usable values or calculates for engine RPM, torque, power, etc. Just a tractive force, some drag forces, the car's mass and a friction coefficient.

But I'm still not clear - assuming for a second that I did have a model that relates engine power to tractive force, or could build one, how would that help me to build a model for resitive forces of different surface types at different speeds?

20. Feb 2, 2016

### A.T.

As already said, this is what usually limits speed for normal cars on a dry road.

Just make sure tractive force is never more than nomal force * static friction coefficient, unless you want to simulate sliding.