For the normal force in the drawing to have the same magnitude at all points on the vertical track, the stunt driver must adjust the speed to be different at different points. Suppose, for example, that the track has a radius of 2.06 m and that the driver goes past point 1 at the bottom with a speed of 24.6 m/s. What speed must she have at point 3, so that the normal force at the top has the same magnitude as it did at the bottom?

For the normal force in the figure to have the same magnitude at all points on the vertical track, the stunt driver must adjust the speed to be different at different points. Suppose, for example, that the track has a radius of 3.1 m and that the driver goes past point 1 at the bottom with a speed of 17 m/s. What speed must she have at point 3, so that the normal force at the top has the same magnitude as it did at the bottom?

To solve this problem, we need to use the principle of conservation of mechanical energy. At the bottom of the track, the total mechanical energy is the sum of the kinetic energy and the gravitational potential energy:

E1 = KE1 + PE1

Where E1 is the total mechanical energy at point 1, KE1 is the kinetic energy at point 1, and PE1 is the potential energy at point 1.

Since the speed at point 1 is provided as 24.6 m/s, we can calculate the kinetic energy at point 1 using the formula:

KE1 = 0.5 * mass * velocity^2

Next, let's consider the mechanical energy at the top of the track (point 3). The kinetic energy at the top is given by:

KE3 = 0.5 * mass * velocity^2,

where we need to find the velocity at point 3.

At the top of the track, the potential energy is given by:

PE3 = mass * g * height,

where g is the acceleration due to gravity (9.8 m/s^2) and height is the difference in height from point 1 to point 3.

Since the total mechanical energy is conserved, we have the equation:

E1 = E3

KE1 + PE1 = KE3 + PE3

Now, let's substitute the expressions for KE1, KE3, PE1, and PE3.

0.5 * mass * velocity1^2 + mass * g * height = 0.5 * mass * velocity3^2 + mass * g * 2 * radius

Notice that we use 2 * radius for the height since the difference in height between point 1 and point 3 is equal to 2 times the radius of the track. This is because point 3 is at the top of the track and point 1 is at the bottom.

Next, we can cancel out the mass from both sides of the equation and simplify:

0.5 * velocity1^2 + g * height = 0.5 * velocity3^2 + 2 * g * radius

Now, let's solve for velocity3:

0.5 * velocity3^2 = 0.5 * velocity1^2 + g * (height - 2 * radius)

Finally, we can take the square root of both sides of the equation to find the value of velocity3:

velocity3 = sqrt(velocity1^2 + 2 * g * (height - 2 * radius))

Substituting the given values of velocity1, g, height, and radius into the equation will give us the speed the stunt driver must have at point 3 for the normal force at the top to have the same magnitude as it did at the bottom.