OldYat47 said:
You won't with power, you will with torque. Rear axle torque is engine torque times reduction ratio. Knowing the radius of the drive wheels gives you drive wheel torque. Now you have force and mass so you can directly calculate acceleration.
You can estimate 1/4 mile times, but that's just an estimate. And restating, you must convert power to force in order to calculate acceleration. I'd be interested in any equations which get from power and mass to acceleration directly. Please post them if you have one or more.
Lastly, power is an "invented" concept, not a fundamental property. James Watt came up with the concept in order to compare water lift capacities of different types of pumps. You can do any and all engineering tasks without ever using power.
Let's take a vehicle powered with an engine (any type) that is transmitting its power through the wheels. What is the maximum force transmitted by the wheels, assuming no friction limit?
First, what do we know about this vehicle? We know it has an engine that produces power and that power cannot be unlimited: It has a maximum power (All engine/motor are defined by their power). We also know that the velocity of the car will change. Now, thanks to the concept of power (Thank you, Mr. Watt!), we can find the maximum force the engine can produce at the wheels:
F_{max} = \frac{P_{max}}{v}
Which gives in graph form:
As you can see, we now know the maximum tractive force the car can produce at any speed, no matter the radius of the wheels, the gear ratios (if any), the engine torque or RPM, even the engine's type (piston, turbine, electric motor, etc.).
As you already mentioned, this available force can be translated into acceleration, if you know the vehicle mass (which is also a fundamental value of your vehicle).
The only thing left to the vehicle designer is to select the appropriate method to make sure that the maximum power will be available at any speed (For example, using a gearbox with a piston engine like in the figure below. Note that - even though each curve correspond to the torque curve of the engine - the «Constant power» curve meets the maximum power in each gear).
From that simple concept of power, I can determine what type of engine will be needed to achieve the acceleration curve I desired, and the fundamental value of the engine will be its power.
So I can choose a diesel engine, do a first draft, find out it pollutes too much, and then decide to change to four electric motors. If I expect the same performance, what will be the common point between the diesel engine and the four electric motors? The sum of the power produced by the four electric motors will be equal to the power produced by the diesel engine.
As for the equations to estimate the ¼-mile, it is an estimate based on a real physics base, not just a statistical curve fitting. The energy gain by the vehicle from 0 to v must be equal to the one delivered by the engine at constant power P during the time t or:
\frac{1}{2}mv^2 = \int_0^t Pdt = Pt
or (equation 1):
v = \left(\frac{2Pt}{m}\right)^{\frac{1}{2}}
Also, the distance d traveled by the car during time t is easily found by integration:
d = \int_0^t vdt = \int_0^t \left(\frac{2Pt}{m}\right)^{\frac{1}{2}}dt = \frac{2}{3}\left(\frac{2P}{m}\right)^{\frac{1}{2}}t^{\frac{3}{2}}
or (equation 2):
t=\left(\frac{3}{2}\left(\frac{m}{2P}\right)^{\frac{1}{2}}d\right)^{\frac{2}{3}}=\left(\frac{3}{2}d\right)^{\frac{2}{3}}\left(\frac{m}{2P}\right)^{\frac{1}{3}}
Since we know that for a ¼-mile, d = 402.336 m (SI unit), then:
t=56.68\left(\frac{m}{P}\right)^{\frac{1}{3}}
or, converting from SI unit to hp and lb:
t_{\lbrack s]}=4.802\left(\frac{m_{\lbrack lb]}}{P_{\lbrack hp]}}\right)^{\frac{1}{3}}
And combining equation 2 with equation 1:
v = \left(3d\frac{P}{m}\right)^{\frac{1}{3}}
Since we know that for a ¼-mile, d = 402.336 m (SI unit), then:
v = 10.65\left(\frac{P}{m}\right)^{\frac{1}{3}}
Converting from SI unit to mph, hp and lb, you get:
v_{\lbrack mph]} = 281.2\left(\frac{P_{\lbrack hp]}}{m_{\lbrack lb]}}\right)^{\frac{1}{3}}
The only thing that vary slightly from the actual equations used across the web are the constants, such that it takes into account some other variables of minor importance. (ref.:
http://stealth316.com/2-calc-hp-et-mph.htm)