An idealized track

__Now that we know how fast an idealized car can go, let's let it run down an idealized track. A typical pinewood derby track is 32 feet long with 4 feet for the starting gate and arresting gear, leaving 8.534 m for travel. The first part of the track has a slope of about 20°, a curved transition section, followed by a flat section to the finish. The track can be approximated by a straight ramp section with length d₁ followed by a flat section with length d₂. Times calculated using the straight track approximation differ by less than 0.001 s compared to those obtained using a curved track and numerical integration.^[2] With the straight track and 20° slope, d₁ = 3.564 m and d₂ = 4.970 m. The car travels down the sloped section d₁ starting at zero velocity and accelerating to a velocity with an average speed of . It then travels the distance d₂ at a velocity . The total time is then

or 12.098/v seconds, if the velocity is specified in meters per second. Run times under different conditions are given in the table below.

Conditions
Frictionless Block	1.219	any	0	4.89	2.47
Stock Wheels	1.219	141.7	3.6	4.75	2.55
Light (4 oz) Car	1.219	113.4	3.6	4.72	2.56
Light Wheels	1.219	141.7	1.0	4.85	2.49
One Raised Wheel	1.219	141.7	2.7	4.78	2.53
Light Wheels, One Raised	1.219	141.7	0.75	4.86	2.49
Rear Bias (Stock Wheels)	1.243	141.7	3.6	4.80	2.52

Center of Mass

__Up to this point, we have approximated the car as a point mass that starts 1.219 m above the finish and winds up at 0 m. As anyone familiar with pinewood derby knows, it is usually best to keep the center of mass as far back as possible, in part to increase the gravitational potential energy. How much faster is a car with the center of mass shifted to the back? Consider two cars, Car A with the center of mass at the center of the block and Car B with the center of mass 25.4 mm (one inch) in front of the rear axle in an extended wheelbase configuration with the axle 15.9 mm (5/8") from the rear of the block. The center of mass of the Car B is 47.6 mm behind Car A and 23.8 mm higher on a 30° slope. Car A starts at 1.219 m and (as shown above) has a maximum velocity of 4.75 m/s and 2.55 s time; Car B starts at 1.243 m and has a maximum velocity of 4.80 m/s and a 2.52 s time.