On a banked race track, the smallest circular path on which cars can move has a radius of 105 m, while the largest has a radius of 175 m, as the drawing illustrates. The height of the outer wall is 18.7 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

To find the smallest and largest speed at which cars can move on the track without relying on friction, we can use the concept of centripetal force.

The centripetal force required to keep an object moving in a circular path is provided by the normal force (perpendicular to the surface) acting on the object. On a banked race track, the normal force can be resolved into two components: the vertical component (opposes gravity) and the horizontal component (provides the centripetal force).

Let's start with finding the smallest speed (a) at which cars can move on the track:

(a) Smallest speed:
To find the smallest speed, we need to consider the scenario where the car is moving on the smallest circular path, with a radius of 105 m. At this radius, the normal force needs to provide both the vertical and horizontal components.

1. Vertical Component:
The vertical component of the normal force is equal to the weight of the car (mg), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, the vertical component of the normal force is given by: N_vertical = mg.

2. Horizontal Component:
The horizontal component of the normal force provides the centripetal force required to keep the car moving in a circular path. As the track is banked, the component of the normal force acting horizontally is directed toward the center of the circle.
We can use trigonometry to find this component. The angle of the banked track can be obtained using tangent:
tan(θ) = height / radius,
where θ is the angle of the banked track and the height is given as 18.7 m.
Therefore, θ = tan^(-1)(18.7 / 105).

The horizontal component of the normal force is given by: N_horizontal = N * cos(θ),
where N is the normal force acting on the car.

Now, equating the horizontal component to the centripetal force:
N_horizontal = (mv^2) / R,
where m is the mass of the car, v is the speed of the car, and R is the radius of the circular path.

Combining the equations above, we get:
N * cos(θ) = (mv^2) / R.

Since we are interested in finding the smallest speed without relying on friction, we assume that the static friction force is zero and the normal force is equal to the weight of the car:

mg * cos(θ) = (mv^2) / R.

Simplifying the equation, we can solve for v:
v = √(g * R * cos(θ)).

Substituting the given values (g = 9.8 m/s^2, R = 105 m, and θ = tan^(-1)(18.7 / 105)), we can calculate the smallest speed.

(b) Largest speed:
To find the largest speed, we repeat the same process as above but for the largest circular path, with a radius of 175 m.

Using the same equation:
v = √(g * R * cos(θ)).

Substituting the given values (g = 9.8 m/s^2, R = 175 m, and θ = tan^(-1)(18.7 / 175)), we can calculate the largest speed.

Note: Make sure to convert angles to radians when using trigonometric functions.