On a banked race track, the smallest circular path on which cars can move has a radius of 113 m, while the largest has a radius of 160 m, as the drawing illustrates. The height of the outer wall is 14.4 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 speeds at which cars can move on the banked race track without relying on friction, we can use the concept of centripetal force.

(a) Smallest Speed:

The smallest speed is achieved when the car is at the bottom of the banked curve. At this point, the weight of the car provides the necessary centripetal force.

Using the formula for centripetal force (Fc = mv^2 / r), we can equate the weight (mg) to the centripetal force:

mg = mv^2 / r

Simplifying and canceling out the mass (m):

g = v^2 / r

Rearranging the equation to solve for velocity (v):

v = √(g * r)

Now we can plug in the values:

v = √(9.8 m/s^2 * 113 m)

Calculating the value:

v ≈ 34.49 m/s

Therefore, the smallest speed at which cars can move on this track without relying on friction is approximately 34.49 m/s.

(b) Largest Speed:

The largest speed is achieved when the car is at the top of the banked curve. At this point, the normal force from the road provides the necessary centripetal force.

Using similar steps as before, we equate the vertical component of the normal force (Nsinθ) to the weight of the car (mg):

Nsinθ = mg

Since the normal force N is perpendicular to the surface of the track, it acts as the centripetal force:

Nsinθ = mv^2 / r

Canceling out the mass (m) and rearranging the equation to solve for velocity (v):

v = √(N * r / sinθ)

As we know the radius (r) and the height of the outer wall (h), we can use trigonometry to find the angle (θ):

sinθ = h / r

Now we can plug in the values and calculate the largest speed:

v = √(N * r / (h / r))

v = √(g * r * r / h)

v = √(9.8 m/s^2 * 160 m * 160 m / 14.4 m)

Calculating the value:

v ≈ 75.88 m/s

Therefore, the largest speed at which cars can move on this track without relying on friction is approximately 75.88 m/s.

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

Let's start with the smallest circular path. The radius of this path is given as 113 m.

(a) Finding the smallest speed:
To determine the minimum speed at which the car can move on this track without relying on friction, we need to consider the forces acting on the car. In this case, the two main forces are the gravitational force (mg) and the normal force (N) exerted by the track on the car.

The gravitational force can be decomposed into two components: one perpendicular to the track (mgcosθ) and the other parallel to the track (mgsinθ).

Since there is no friction, there is no horizontal force acting on the car. Therefore, the net force acting on the car in the horizontal direction is zero. This means that the horizontal component of the gravitational force is canceled out by the horizontal component of the normal force.

Now, consider the vertical forces acting on the car. The vertical component of the gravitational force (mgsinθ) provides the centripetal force required to keep the car moving in a circular path. By equating this force to the centripetal force, we can calculate the minimum speed.

Centripetal force = mgsinθ

In this case, the centripetal force is provided by the vertical component of the gravitational force.

mgsinθ = mv^2 / r

Here, m is the mass of the car, g is the acceleration due to gravity, θ is the angle of banking, v is the speed of the car, and r is the radius of the smallest circular path.

Substituting the given values:

113 m * g * sinθ = m * v^2 / 113 m

Simplifying the equation, we find:

v^2 = 113 m * g * sinθ

Finally, we can solve for v:

v = √(113 m * g * sinθ)

Substituting the value of g (approximately 9.8 m/s²) and the angle of banking (θ), you can calculate the minimum speed at which cars can move on the smallest circular path without relying on friction.

(b) Finding the largest speed:
We can follow the same procedure to find the largest speed. Here, we will use the radius of the largest circular path (160 m) instead of the smallest one.

Using the same formula as above:

v = √(160 m * g * sinθ)

Substituting the value of g (approximately 9.8 m/s²) and the angle of banking (θ), you can calculate the maximum speed at which cars can move on the largest circular path without relying on friction.

Remember to convert the angle of banking to radians if it is given in degrees.

That's how you can find the smallest and largest speeds at which cars can move on the banked race track without relying on friction.