Analysis:

To calculate the angle made by the cycle with the vertical direction, we can use trigonometry. Let's break down the problem into smaller steps and calculate each step step-by-step.

Step 1: Calculate the distance traveled by the cyclist
The length of the circular track is given as 500m. This means that the cyclist completes one full lap of the track in 500m. Since the cyclist completes the track in 50s, we can calculate the distance traveled by the cyclist in 1 second using the formula:

Distance = Total Length / Time Taken
Distance = 500m / 50s
Distance = 10m/s

Step 2: Calculate the horizontal and vertical components of the distance traveled
Since the cyclist is moving in a circular path, we can consider the horizontal and vertical components of the distance traveled. The horizontal component represents the distance traveled along the track, and the vertical component represents the change in height.

Step 3: Calculate the angle made by the cycle with the vertical direction
To calculate the angle, we can use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the change in height and the adjacent side is the distance traveled along the track.

Angle = tan^(-1) (Opposite / Adjacent)
Angle = tan^(-1) (Vertical Component / Horizontal Component)

Step 4: Calculate the values of the vertical and horizontal components
We can use the Pythagorean theorem to calculate the values of the vertical and horizontal components. The hypotenuse is the distance traveled, the horizontal component is the adjacent side, and the vertical component is the opposite side.

Using the formula:

Distance^2 = Horizontal Component^2 + Vertical Component^2

We can rearrange the formula to solve for the vertical component:

Vertical Component = √(Distance^2 - Horizontal Component^2)

Step 5: Substitute the values into the angle formula
Now that we have calculated the values of the vertical and horizontal components, we can substitute them into the angle formula to find the angle made by the cycle with the vertical direction.

Angle = tan^(-1) (Vertical Component / Horizontal Component)

Conclusion:

By following the above steps, we can calculate the angle made by the cycle with the vertical direction.