The Indianapolis Motor Speedway has four banked curves, each of which forms a quarter of a circle. Suppose a race car speeds along one of these curves with a constant tangential speed of 75.0 m/s. Neglecting the effects due to the banking of the curve, the centripetal acceleration on the car is 22.0 m/s2. What is the radius of the curve?

To find the radius of the curve, we can use the formula for centripetal acceleration which is given by: \[ a_c = \frac{v^2}{r} \] Here, \( a_c \) is the centripetal acceleration, \( v \) is the tangential speed of the car, and \( r \) is the radius of the curve.

We are given: \[ a_c = 22.0 \, \text{m/s}^2 \] \[ v = 75.0 \, \text{m/s} \]

Rearrange the formula to solve for the radius: \[ r = \frac{v^2}{a_c} \]

Substitute the given values: \[ r = \frac{(75.0 \, \text{m/s})^2}{22.0 \, \text{m/s}^2} \]

Calculate the value: \[ r = \frac{5625 \, \text{m}^2/\text{s}^2}{22.0 \, \text{m/s}^2} \] \[ r \approx 255.68 \, \text{m} \]

So the radius of the curve is approximately 255.68 meters.

The concept in play here is centripetal force and centripetal acceleration. When an object moves in a circular path, it experiences an inward force called centripetal force, which is responsible for changing the direction of the object's velocity, causing it to move in a circular motion. This inward force is not an independent force but rather a result of other forces, like gravity, friction, or in the case of a car on a track, the frictional force between the tires and the road surface.