The contestants at a cycling race travel around a 0.5km radius track. The peloton travels 33 laps in 1 hour to

complete the race. What is the total angular distance they have traveled? On the first lap, they start from rest
and after traveling ½ the circumference of the course, they are traveling at 50 km/h. Assuming a constant
acceleration of the peloton, how long did it take them to reach that point?

To find the total angular distance the contestants have traveled, we need to determine the circumference of the track and the number of laps they have completed.

1) Circumference of the track:
The circumference of a circle with a radius of 0.5 km can be calculated using the formula C = 2πr, where C represents the circumference and r represents the radius.
C = 2π(0.5 km)
C ≈ 3.14 km

2) Number of laps completed:
Given that the peloton travels 33 laps in 1 hour, we know that they complete the race in that time. Therefore, the number of laps they have completed is 33.

To find the total angular distance, we multiply the circumference of the track by the number of laps completed:
Total angular distance = Circumference × Laps completed
Total angular distance = 3.14 km × 33
Total angular distance ≈ 103.62 km

Now, let's move on to the second part of the question, which asks how long it took the peloton to reach a speed of 50 km/h after traveling half the circumference of the course.

1) Distance traveled:
To find the distance traveled on the first lap, we need to calculate half the circumference of the course since they start from rest and reach this point.
Distance traveled = (1/2) × Circumference
Distance traveled = (1/2) × 3.14 km
Distance traveled ≈ 1.57 km

2) Acceleration:
Given that the acceleration of the peloton is constant, we can use the formula v^2 = u^2 + 2as to find the time it took to reach a speed of 50 km/h.

- v represents the final velocity (50 km/h)
- u represents the initial velocity (0 km/h, as they start from rest)
- a represents the acceleration (unknown)
- s represents the distance traveled (1.57 km)

Rearranging the formula, we get:
a = (v^2 - u^2) / (2s)

Plugging in the values:
a = (50^2 - 0^2) / (2 × 1.57)
a = 2500 / 3.14
a ≈ 796.18 km/h^2 (approximated)

Now, we can use the formula v = u + at to find the time it took to reach a speed of 50 km/h.

- v represents the final velocity (50 km/h)
- u represents the initial velocity (0 km/h)
- a represents the acceleration (796.18 km/h^2)
- t represents the time (unknown)

Rearranging the formula, we get:
t = (v - u) / a

Plugging in the values:
t = (50 - 0) / 796.18
t ≈ 0.0627 hours (approximated)

To convert the time into minutes, we multiply by 60:
t ≈ 0.0627 hours × 60 minutes/hour
t ≈ 3.76 minutes (approximated)

Therefore, it took the peloton approximately 3.76 minutes to reach a speed of 50 km/h after traveling half the circumference of the course.