MadSci Network: Engineering

Re: horsepower

Date: Tue May 19 01:47:10 1998
Posted By: Don Pettibone, Other (pls. specify below), Ph.D. in Applied Physics, Quadlux Inc.
Area of science: Engineering
ID: 894329836.Eg

Good question.  You are right that there is more to how fast a car goes 
than just knowing the horsepower rating of the engine.

First, some basics.  Horsepower is a unit of power.  Power has units of 
energy/time (energy per unit time).  Physicists use a unit called a joule 
to measure energy in a system of units known as MKS (Meters Kilograms 
Seconds, as the units for length, mass and time, respectively).  A watt is 
a unit of power that equals one joule per second.   One horsepower equals 
about 746 watts.  James Watt came up with the unit of horsepower by 
measuring how much power a horse could deliver.  [Note that he did not name 
the watt after himself, that would have been unseemly.  If you are a really 
good physicist you hope that someone else names a unit after you, probably 
well after you are dead.]  The Encyclopedia Britannica says that James Watt 
must have measured a really strong horse, or maybe he only measured how 
much power it could put out for a short time.  A more reasonable number for 
a horse working an 8 hour day is about 500 watts.  You can get a rough idea 
of how much power a watt is by knowing that a bright light bulb draws about 
100 watts, and that a typical microwave oven draws about 1000 watts.  A 
person who is reasonably fit can put out about 200 watts for times of the 
order of an hour or more.  I believe, though I couldn't confirm this 
anywhere, that the American cyclist Lance Armstrong has been clocked at 500 
watts power output on a stationary bike, but I don't know for how long.  
You could say that he is as strong as a (typical) horse.

Horsepower rating and how fast cars go is pretty complicated.  I will 
simplify things a bit to try and get the essential physics right, and in so 
doing I will gloss over some important points.  First, let's make a 
distinction between top speed and acceleration.  You probably have noticed 
that large tractor trailer rigs are very slow in reaching their top speed; 
their acceleration is low.  This is due to the fact that they are very 
massive.  However, once you get that mass going at 70 miles per hour you do 
not need to add any more energy to maintain that velocity, if it wasn't for 
friction.  Friction for a car or a truck is largely due to rolling friction 
plus air resistance.  Neither of these depend directly on mass, so a truck 
that is empty has a top speed very close to that of a truck that is loaded 
up.  [This is not quite true, as the rolling friction is probably a 
function of how much the tires are deformed by the load, with higher 
rolling resistance matching larger tire deformations, corresponding to a 
heavy load.]  So top speed depends just on horsepower and the drag on the 

Now let's consider acceleration.  Acceleration has units of velocity/time 
(velocity per unit time).  A car that is accelerating at 10 meters per 
second every second is going faster and faster.  If it starts from rest, 
then after one second it is going 10 meters/second, and at the end of two 
seconds it is going 20 meters/second, and so on.  Newton's second law helps 
us out in determining how fast a car can accelerate.  To make things 
simple, we will consider acceleration in the absence of any friction.  Then 
Newton's 2nd law says that force equals mass times acceleration.  So 
acceleration is just the force divided by the mass.  So mass counts; the 
more massive a vehicle is, for the same power, the more slowly it will 
accelerate.  We will run through some typical numbers to see how things 
work out for a typical car.  A car that weighs about 3400 pounds (= 1550 
kilograms) and has a horsepower rating of 200 horsepower (149,000 watts) 
can accelerate to 60 miles per hour (26.8 meters/second) in about 8 
seconds.   Now if all the energy of the engine went into kinetic energy of 
the vehicle, we could see how the numbers checked, since we know that the 
product of the power and the time equals the energy the engine has put out, 
and we know the kinetic energy of the vehicle equals one half the mass 
times the velocity squared.
Engine energy =	Power*time=		149,000*8=1,2000,000 joules.
Kinetic Energy=	0.5*1550*26.8*26.8=		   560,000 joules.
What went wrong?  You will notice that a little less than half of the 
energy showed up as kinetic energy.  Where did it all go?  Im not sure 
where all of it went, but I will hazard guesses as to where most of it 
went.  An engine that is rated at 200 horsepower only puts out 200 
horsepower near the top end of its R.P.M range (revolutions per minute of 
the motor).  A car that rapidly accelerates to 60 miles per hour will spend 
a lot of time at lower R.P.M.s than this, as the cars gears are shifted 
from low gear to successively higher gears.  I think that accounts for most 
of the discrepancy.  Two other factors worth mentioning are that not all 
the energy that the motor puts out goes into translational kinetic energy 
of the car, a fairly large fraction is tied up in rotational kinetic energy 
of the car engine, the drive train and the wheels.  I had a physics 
professor who gave a talk once on The Physics of Drag Racing, and I think 
he put the number at something of the order of 20% for a racing stock car.  
That is, the ratio of rotational kinetic energy to translational kinetic 
energy is about 0.2.   Another factor we have ignored is friction.  At low 
speeds friction is not a major factor, but certainly above 30 miles per 
hour we should be taking it into account.

I have not done any of these calculations using torque, so I'll just 
mention briefly what it is.  Torque times angular velocity equals the power 
output, so if you know how fast the engine is turning over and you know the 
torque you also know the power output.  For a given wheel radius, torque is 
proportional to the force the wheels can exert on the ground, so the torque 
divided by the mass of the vehicle is proportional to the acceleration.  
Torque, like horsepower, is also a function of engine speed, with the 
available torque dropping off at low engine RPM.

So you are correct in thinking that horsepower is not the only important 
thing in determining how fast a car or truck goes.  For maximum 
acceleration you need a lot of power in a light vehicle and a light drive 
train.  You also need a system of gears that allows you to get the most 
energy out of the engine at both high and low speeds.  For high speed you 
need a lot of power and not much friction, which means a streamlined shape.

I hope this helps.  Drive safely.

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