Horsepower is Torque

Recapping from Horsepower - Part 2, the force pushing the car forwards (F) comes from the engine torque (T=500 ft-lb) multiplied by the gear ratio (gr=15) to get wheel torque (T*gr=7500 ft-lb). The wheel torque is then converted to pushing force (F=6667lb) at the tire contact spot, which pushes with a force that is larger the shorter the "lever arm" (the tire radius, tr=13.5"=1.125').

So,

• F = T*gr/tr
• (see my little tutorial on Math if you are freaked out by this equation!)

Power (P) is Work (W) per unit time, and Work is Force (F) applied over a distance (d).

• P = W / t
• W = F * d

so

• P = F * d / t

But a distance per unit time (d/t) is speed (S), so another way of saying this is that Power (P) is a Force (F) consistently applied at Speed (S)

• P = F * S

So if I'm able to continue applying a force of say 1167 lb to the wheels while travelling at 180 mph, that corresponds to 560 HP.

• 180 mph = 180 * 5280 feet per hour = 180*5280/60 feet per minute
• Power = 1167 lb * (180*5280/60) ft/min =18,485,280 ft-lb/min
• 1 HP = 33,000 ft-lb/min
• Power = (18,485,280/33,000) HP = 560 HP

If we can apply that 1167 lb while still moving that quickly, well that's power my friend! At that speed, the air starts pushing back with that same force of 1167 lb, and that defines the top speed you can go, when the force pushing you forward is the same as the force holding you back, then you won't accelerate or decelerate, you'll just strain against the wind at top speed.

Another way of looking at it is say I am in a tiny old Renault Cinq (my first car, thank you very much!) and I have it running at 60 mph on the highway.

If the most force I can muster in top gear with the engine running so fast is say 500 lb, then my little car is only generating 80 HP. Say I saved my pennies and installed a sporty new exhaust which cranked my horsepower to 100 HP at those same engine revs. Then at that same 60 mph speed I can now generate 620 lb of force, which is an excess of 120 lb over the 500 lb of air resistance that holds me back at that speed, which means I can speed up again past 60 mph, maybe even all the way to 70 mph!

So power and top speed are very related. Power is torque delivered at speed.

So how is speed (S) related to engine RPMs? Speed is distance per unit time. RPMs is rotations per unit time. So we have the time part, but how do rotations relate to distance?

If the engine rotates at a certain RPM, then the tires rotate slower due to the gear ratio, gr times slower in fact. So say the M5 is making 3000 RPMs at the crankshaft. In first gear that translates into 1/15th of that at the wheels, so the wheels are only spinning 200 times per minute.

The distance travelled each minute is the number of times the wheels rotate around each minute, times the perimeter around the edge of the tires.

In the picture above, if the wheels rotate around once, the car would go the distance indicated, equivalent to once around the perimeter of the tires.

• Speed = (wheel rotations per minute) * (tire perimeter)

The wheel rotations are RPM/gr. The tire perimeter is PI (3.14) times the tire diameter, which is twice the tire radius, so PI*2*tr. Bringing this and the other two relations all together we get

• S = (RPM/gr) * (PI*2*tr)
• F = T*gr/tr
• P = F * S

So power is proportional to force and speed. Force goes up with gear ratio and down with tire radius. Speed goes down with gear ratio and up with tire radius in exactly the same proportions. What that means is that the tire radius and the gear ratio both get cancelled out, leaving

• P = T * RPM * PI*2

Very cool! If T is in ft-lb and RPM in revolutions per minute, then P is in units of ft-lb/min. Recall that Watt's draft horse could do 33,000 ft-lb/min, so the P above needs to get divided by 33,000 to get to Horsepower.

• P = T * RPM * 2*PI/33000 

Or, more simply,

• P = T * RPM / 5252 

So we can see that the torque and power curves are really sort of the same thing. If we know the torque at any RPM, we know the power of the car at that RPM as well. Remember that power is force at speed. Torque is "circular force" and RPMs are "circular speed", so it's natural that power is circular force times circular speed, as we showed above.