# Speed loss during gear change

1. Apr 13, 2010

### ajwillenbrook

Hey

I have a assignment to produce a excel work book to calculate acceleration time and top speeds for different gear ratios in race cars. I have so far work out time to accelerate to top speed, but I want to add a bit more detail to my graphs by showing the speed loss during gear changes.

car mass - 600kg
drag coefficient - 1.3
density air - 1.225kg/m^3
frontal area - 0.4m^2
rolling resistance = 125N
shift time - 0.5 seconds

I can work out the momentum and drag forces at the point the clutch is pressed at the start of the gear change using

momentum = mass * velocity
= 600v

drag force = (0.5 * coefficient drag * density air * frontal area * velocity^2)+ rolling resistance
= 125+0.3185v^2

Sure it can be work out using calculus but has been many years since I last did that.

Any help would be greatly appreciated

Thanks Andy

2. Apr 15, 2010

### Filip Larsen

I'm not sure why you model rolling friction as a constant force. If that is really what you want to do, perhaps you can find some formulas describing speed as function of time in https://www.physicsforums.com/showthread.php?t=393048

Normally, though, one would model rolling friction as proportional to speed, that is, your equation of motion during gear shift would be

$$m \dot{v} = -k v^2 - c v$$

where $m, k, c > 0$. The c constant is the one corresponding to your 125N (at some speed?) If I quickly integrate this (assuming this is not really part of your homework, and if it is, you should figure out to do this integral yourself), I get

$$v = \frac{d}{(1 + d/v_0)\exp(ct/m)-1}$$

where v0 is the initial speed at t = 0 and $d = c/k$.