A car drives straight down toward the bottom of a valley and up the other side on a road whose bottom has a radius of curvature of 110m . At the very bottom, the normal force on the driver is twice his weight.

At what speed was the car traveling?

To find the speed at which the car was traveling, we can use the concept of centripetal acceleration and the relationship between forces in circular motion.

Let's break down the problem and go step by step:

Step 1: Draw a free-body diagram of the car at the bottom of the valley.

There are two forces acting on the car: the weight (mg) pointing downward and the normal force (N) pointing upward. At the bottom of the valley, the normal force is twice the weight, so we can write N = 2mg.

Step 2: Identify the net force acting on the car at the bottom of the valley.

The net force is the vector sum of all the forces acting on the car. In this case, the net force can be determined by subtracting the weight from the normal force:
Net force = N - mg = 2mg - mg = mg.

Step 3: Determine the centripetal force at the bottom of the valley.

In circular motion, the centripetal force is responsible for keeping the object moving in a curved path. In this case, the centripetal force is provided by the net force. Therefore, we can write:
Centripetal force = mg.

Step 4: Apply the centripetal force formula.

The centripetal force (Fc) can be calculated as the product of mass (m), velocity (v), and the radius of curvature (r):
Fc = mv^2 / r.

In this case, the radius of curvature (r) is given as 110m, and we already determined the centripetal force as mg. Combining these, we get:
mg = mv^2 / r.

Step 5: Solve for the speed (v).

Rearranging the formula, we have:
v^2 = rg.

Substituting the given values, we get:
v^2 = 110m * 9.8 m/s^2.

Simplifying:
v^2 ≈ 1078 m^2/s^2.

Finally, taking the square root of both sides, we find:
v ≈ 32.85 m/s.

Therefore, the car was traveling at approximately 32.85 m/s.

A car drives straight down toward the bottom of a valley and up the other side on a road whose bottom has a radius of curvature of 100m . At the very bottom, the normal force on the driver is twice his weight.

At what speed was the car traveling?