Power to acceleration relationships

No maths to back any of this up, which really shows because most of them are wrong.

Been looking at some stats of cars with different engines, and same kerb weight (give or take) and I have come to the following conclusions: 1) Acceleration (call it 0-60) is proportional to the square of horsepower (i.e. a 100 hp car that does 0-60 in 10 seconds, needs 400 hp to do it in 5 seconds).

2) Acceleration is proportional to the square root of weight. Case being a BMW 320d does 0-60 in 7.3 sec and top speed 146 mph at a weight of 1420kg, whilst the 520d does 7.6 sec and 143 mph and 1620 kg

3) Weight has no/ miniscule effect on top speed. Case being the BMW comparison above, and the 5 series is probably less aerodynamic than the 3 series because it's wider.

3) Fuel economy is proportional to the square root of car weight. I.e. if you were to quadruple a cars mass, its fuel economy would half

4) And finally, because drag force is proportional to the cube of velocity, it must take 2^3=8 times longer for a car to hit 50 mph compared to 100 mph

No maths to back this up.


Acceleration is not proportional to the square root of the weight. It will have some form of inverse proportionality, but not to the square root.


Wrong. The 5 series has a more powerful engine, and thus negates the extra weight.


Again, no maths to back this up.


"it takes 8 times longer for a car to reach 50mph than 100mph". Just think about that statement, if you can't figure out why it's wrong then God help you.

If you think a BMW 520d has more power than a 320d then god help you.

So what do you think all the relationships are ? Looks to me like you're only capable of picking holes in other peoples arguments and not creating a counter agument of your own.

Fair enough, the cars have the same engine, same power.

First of all, I told you that acceleration is inversely proportional to the weight. The fact is, the maths of your arguments don't add up and if you had any concepts of mathematics you'd know that.

4) And finally, because drag force is proportional to the cube of velocity, it must take 2^3=8 times longer for a car to hit 50 mph compared to 100 mph

Wrong.

Drag is proportional to the square of velocity, not the cube.

$[br]D = 0.5 \rho V^2 A C_d[br]$

Source: just about any aerodynamics book. But if you don't have one to hand: http://en.wikipedia.org/wiki/Drag_coefficient

8 times longer for a car to hit 50, than 100mph... da*** lol?

does the wind add power to your Passat then, hurricane force winds past 50 mph I bet

A couple to start you off: force = mass x acceleration. Rearrange and you get acceleration = force/mass, so acceleration is inversely proportional to weight. No squares or square roots.
You're correct in saying that weight has little to no impact on top speed, with the proviso that you have a sufficiently long time to get to top speed (obviously the heavier car will take longer to reach its top speed). Top speed is reached when the drag forces on the car exactly balance the driving force from the car.
Fuel economy is a complex one, as it depends on what type of driving you're doing. At steady speed on a flat, straight road the only factor is drag, so weight doesn't have any influence. Drive a twisty back road and it makes a big difference.

As for your other statements, you can't really extrapolate them from a car spec sheet. For example, an engine has a rated output power, but it doesn't make that power all the time, so the shape of the power curve is important when comparing engines. You don't know that they're using the same transmissions, so the losses there may be different. It's also not unknown for car manufacturers to slightly derate an engine so the flagship model looks better - there's a rumour that Peugeot turned down the boost on the 1.9 turbo diesel because it was outperforming the 2.0 petrol. The testing may have been done under slightly different conditions or on different equipment. They might have had different wheel/tyre combinations, with different rolling radii. And that's only a few off the top of my head. To get any kind of meaningful results you'd need to run some experiments where you can change just one variable.

So if you had a 1000 kg car that did 0-60 in 10 seconds, and added a 100 kg passenger, the 0-60 would fall to 11 seconds? Not sure I buy that one.

So a car will take 4 times longer (roughly according to gearing etc) to hit 100 than 50mph ?

Err no. How were you even getting that from that equation? The drag will not be the same at 100mph as it was at 50mph. The drag will go up four times (neglecting any chance in density, and assuming the drag coefficient remains constant) if the speed increases from 50mph to 100mph. Therefore you'll (in crude terms) need approximately four times the power to reach 100mph as you do to reach 50mph.