**Introduction**

When considering the motions of automobiles, the calculating
acceleration, velocity, and distance can be increased if the power output of
the vehicle and the vehicle’s mass are considered.

It is necessary to go beyond the simple equations of motion
when considering vehicles. For reference,
I list the simple of equations of motion here:

v = a*t + v0

x = a*t^2/2 + v0*t + x0

where a is acceleration, v is velocity, and x is position. v0 and x0 are initial velocity and initial position,
respectively. Acceleration is assumed to
be constant.

**Considering Power**

In the paper “Constant powers equation of motion”, Roger
Stephenson (see Sources below) explained that the automobile’s engine can
produce constant torque at a certain RPM (revolutions per minute). However, for the car to accelerate requires
either the engine’s rpm or gear ratio linking the engine to the wheels must
change. Drivers aim to change gears to
engine rpm at a level where the most power is produced. This causes torque at the wheels to change. However, power is kept at a constant level
and can be used to modify the equations of motion.

Let P be the power of the vehicle and m it’s mass. The kinetic energy of the automobile at t=0
is:

KE = ½ * m * v^2 = P * t

Solving for velocity, we get

‘v = √(2*P*t/m)

Stephenson introduced the variable Z, zip, defined as:

Z = √(P/m)

Z nicely combines the vehicle’s constant power and
mass. Deriving the equations of motion
started with the recognition of relationship between work and power:

Power = Work / Time

Since Work = Force * Displacement and by Newton’s Second
Law, Force = Mass * Acceleration, we arrive at:

P = W /t

P = m * a * x/t

P/m = a * x/t

Since Z = √(P/m) and v = x/t, and recognizing that a =
dv/dt,

Z^2 = dv/dt * v

From there, Stephenson derived the following equations of
motion under constant motion in Appendix A in his paper:

v = Z * (2*t + (v0/Z)^2) ^ (1/2)

x = (Z/3) * ( (2*t + (v0/Z)^2) ^ (3/2) – (v0/Z)^3) ) + x0

Where Z = √(P/m)

Using standard US units, power is stated in ft*lb/s and mass
is stated in slugs. The following
conversion factors may be necessary:

1 hp (horsepower) = 550 ft*lb/s

1 slug = 32.17404 lbs (mass * gravity)

1 mph = 22/15 ft/s

1 ft/s = 15/22 mph

**The Program CARFORCE**

The following HP Prime program, CARFORCE, takes the
following for arguments:

P = power in horsepower (hp)

M = mass in pounds (lb)

I = initial velocity (mph)

T = time to evaluate (seconds)

CARFORCE will return a list of four answers:

Zip (numerical value only (ft/s^1.5))

Acceleration in miles per hour per second

Velocity in miles per hour

Position in feet

This program illustrates the use of units in an HP Prime
program. A way to separate a value from
a value_unit couple is to divide such value by 1_unit. Example:

15_ft / 1_ft = 15

The program listed is shown below. The program assumes that x0 = 0.

EXPORT CARPOWER(P,M,I,T)

BEGIN

// 2014-06-14 EWS

// Roger Stevenson, 1980

// Lloyd W. Taylor, 1930

// power (hp),mass (lb),

// initial velocity (mph), time (sec)

LOCAL Z,W,A,V,X,L0;

// convert

P:=CONVERT(P*1_hp,1_(ft*lbf/s))/1_(ft*lbf/s);

I:=CONVERT(I*1_mph,1_(ft/s))/(1_(ft/s));

M:=CONVERT(M*1_lb,1_(slug))/(1_slug);

Z:=√(P/M);

// temp

W:=2*T+I²/Z²;

A:=Z/√W;

V:=Z*√W;

X:=Z/3*(W^1.5-(I/Z)^3);

L0:={Z,A*15/22,V*15/22,X};

L0:=L0*{1,1_(mile/(h*s)),1_(mile/h),1_ft};

RETURN L0;

END;

BEGIN

// 2014-07-25 EWS

. use decimal conversions

I:=22/15*I; // mph to ft/s

M:=M/32.1740485564;

L0:={Z,A*15/22,V*15/22,X};

Example:

Data: Vehicle has a
100 hp engine, has a mass of 4,000 pounds.
Figure the equations of motion after 10 seconds elapsed. Assume the vehicle is initially stopped.

CARPOWER(100, 4000, 0, 10) returns

{21.0331445022, 3.2066959696_mile/(h*s),
64.1339193918_mile/h, 627.087211833_ft}

**Sources**

Revised 6/12/2014

Stephenson, Roger “Constant
power equations of motion” (Published
January 29, 1982) from the compilation book

__The Physics of Sports__edited by Angelo Armenti, Jr. American Institute of Physics: New York 1992, pp. 284-289
Considering power, calculating acceleration, velocity, and position become more accurate.

Until next time, have a great day! Happy Birthday Mom and Dad! Happy Father’s Day!

Eddie

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