We want to fill in velocities as we go down the columns, so we need to solve equation (4) for vi. This will give us a formula for computing vi given vi-1 for every row except the first. In the first row, we put the speed with which we enter the straight, which is an input to the problem. We get:

 (5)

We label another column distance, and we call the distance value in the i-th row xi. Just as acceleration is the rate of change of velocity, so velocity is the rate of change of distance over time. Just as before, then, we may write:

 (6)

Solved for xi, this is:

 (7)

Equation (7) gives us a formula for calculating the distance for any time given the previous distance and the velocity calculated by equation (5). Physicists would say that we have a scheme for integrating the equations of motion.

A small detail is missing: what is the force, F? Everything to this point is kinematic. The real modelling starts now with formulas for calculating the force. For this, we will draw on all the previous articles in this series. Let's label another column force, and a few more with drag, rolling resistance, engine torque, engine rpm, wheel rpm, trans gear ratio, drive ratio, wheel torque, and drive force. As you can see, we are going to derive a fairly complete, if not accurate, model of accelerating down the straight. We need a few constants:

CONSTANT	SYMBOL	EXAMPLE VALUE
rear end ratio	R	3.07
density of air		0.0025 slugs / ft3
coeff. of drag	Cd	0.30
frontal area	A	20 ft2
wheel diameter	d	26 in = 2.167 ft
roll resist factor	rr	0.696 lb / (ft / sec)
car mass	m	100 slug
first gear ratio	g1	2.88
second gear ratio	g2	1.91
third gear ratio	g3	1.33
fourth gear ratio	g4	1.00

and a few variables:

VARIABLE	SYMBOL	EXAMPLE
engine torque	TE	330 ft-lbs
drag	Fd	45 lbs
rolling resistance	Fr	54 lbs
engine rpm	E	4000
wheel rpm	W	680
wheel torque	TW	1930 ft-lbs
wheel force	FW	1780 lbs
net force	F	1681 lbs

All the example values are for a late model Corvette. Slugs are the English unit of mass, and 1 slug weighs about 32.1 lbs at sea level (another manifestation of F = ma, with F in lbs, m in slugs, and a being the acceleration of gravity, 32.1 ft/sec2).