Chapter 3.4, Problem 1SP

How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors:

• The weight of the car;
• The friction between the tires and the road;
• The radius of the circle;
• The “steepness” of the turn.

In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this.

A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle $\theta$ . If the height of the car off the ground is h, then the position of the car at time t is given by the function $r\left(t\right)\langle R\cos \left(\omega t\right),R\sin \left(\omega t\right),h\rangle$ .

Figure 3.20 Views of a lace ear moving around a track.

1. Find the velocity function $v\left(t\right)$ of the car. Show that vis tangent to the circular me. This means that, without a force to keep the car on the curve, the car will shoot off of it.