Fast Car Physics

Chapter 2. Horsepower, 0 to 60 mph, and the Quarter Mile

Chapter 2 Horsepower, 0 to 60 mph, and the Quarter Mile 2.1 HORSEPOWER We made it all the way through the discussion in the first chapter without really defining horsepower. Power, P, is the rate of doing work, W/t. Work, W, in physics is a force, F, acting along a distance, d. Combining these definitions , P  W  Fd  Fv. t t This assumes that the force, F, and the velocity, v, are collinear (we’ll worry about vector products later). It is most easily calculated in drag racing as the product of the force in pounds with the velocity in feet per second. It therefore has the units of ft-lbs/s. A more convenient unit is horsepower. One horsepower is equal to 550 ft-lbs/s. It is common practice to refer to the quantity power as horsepower. The force that is doing this work is the force that is accelerating the car, that is, friction with the road surface, FRT : Horsepower  (FRT V)/550. (2.1) horsepower, 0 to 60 mph, and the quarter mile 19 From Newton’s second law FRT = ma, where m is the mass of the car in slugs and a is the acceleration in ft/s2 . For the STi, the mass is 102.2 slugs and the acceleration can be obtained from figure 1.6. With each shift of the gears, the velocity increases and the force available to push the car decreases. It goes from a peak of about 3650 lbs in first gear to a minimum of 640 lbs in sixth gear at 145 mph at 5500 rpm. The STi is electronically limited to 145 mph, but what if it weren’t? There is still 1500 rpm to reach the 7000 rpm redline, in theory, 185 mph. Unfortunately, it will never get there. Horsepower determines the car’s ability to overcome the resistive forces that act on the car. The major factor is the aerodynamic drag. When the rate at which the tires do work is equal to the rate at which drag does work, the car is in equilibrium. At this point, the drag force fdrag  FRT . So, in a sense the horsepower and the drag force determine the maximum speed of the car. A higher gear won’t help raise the top speed because, as we have seen, higher gears actually lower the force available at the tire-road interface. Horsepower cannot be decoupled from torque. They are closely related. From equation (1.2) FRT  W R and from equation (1.7) V  R, where W is the torque of the road acting on the wheel. R is the radius of the tire, V is the speed of the car, and  is the angular velocity in radians per second, which can also be expressed in revolutions per minute. Plugging equations (1.2) and (1.7) into equation (2.1), Horsepower  W /550. (2.2) The 550 is a unit conversion factor. Hence, horsepower is really the product of the torque, rpm, and a couple of constants to get the units correct. They are closely related, and it is easy to see how the enthusiast becomes distracted by the wrong quantity. Torque is directly related to acceleration, and horsepower is related to top speed. 20 fast car physics It is worthwhile to make sure we follow the horsepower units. In introductory physics we have Power (J/s)  (N . m)  (rad/s) or, in British units, Power(ft – lbs/s)  (ft – lbs)  (rad/s). Conversion factors include 1 horsepower  746 watts  550 ft  lbs/s  rad  2 rad  1 min  (rpm). s 1 rev 60s  Using these definitions and conversions, we can relate horsepower, torque in ft-lbs, and  in rpm: (2.3) P(horsepower)  (ft  lbs)  (rpm) 60 550 2 1  P(horsepower)  (ft  lbs)  (rpm) . 5252 When engine rpm is 5252, the torque should equal horsepower. As a quick check for dyno curves from an unknown source, torque and horsepower must cross at 5252 rpm. If they do not, something is wrong. With these relationships in hand and a few simplified models, it is possible to visualize the relationship between torque and horsepower. For example, if the torque is constant and we apply equation (2.3), we get a constantly increasing power as shown in figure 2.1. If the torque increases linearly with rpm, then the power increases parabolically with...