Consider an old-fashion bicycle with a small wheel of radius 0.15 m and a large wheel of radius 0.8 m. Suppose the rider starts at rest, accelerates with a constant acceleration for 2 minutes to a velocity of magnitude 12 m/s. He maintains this velocity for 14 mintues and then decelerates, with a constant deceleration, for 4 minutes at which time he is at rest.
a) Find the total distance traveled by the rider. Give your answer in kilometers.
b)Find the total angles through which the small wheel turned during the trip.
c)Find the angular velocities of the two wheels during the constant velocity part of the trip,
d)Find the angular accelerations of the two wheels during the first two minutes.
a) To find the total distance traveled by the rider, we need to calculate the distances covered during each phase of the trip and then add them together.
1. Acceleration phase: The rider starts from rest and accelerates for 2 minutes to a velocity of 12 m/s. We can use the equation for constant acceleration to calculate the distance traveled during this phase:
Distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)
The initial velocity is 0 m/s, the final velocity is 12 m/s, and the time is 2 minutes (or 120 seconds). The acceleration can be calculated as the change in velocity divided by the time:
Acceleration = (Final velocity - Initial velocity) / Time
Plugging in the values, we can find the distance traveled during the acceleration phase.
2. Constant velocity phase: The rider maintains a velocity of 12 m/s for 14 minutes. The distance traveled during this phase can be found by multiplying the velocity by the time:
Distance = Velocity * Time
3. Deceleration phase: The rider decelerates with a constant deceleration for 4 minutes until coming to rest. The final velocity is 0 m/s, the initial velocity is 12 m/s, and the time is 4 minutes (or 240 seconds). We can use the same equation as in the acceleration phase to find the distance traveled during the deceleration phase.
Once we have the distances covered in each phase, we can add them together to get the total distance traveled.
b) To find the total angles through which the small wheel turned during the trip, we can use the formula:
Angle = Distance / Radius
We already found the distances in part a), and we know the radii of the small and large wheels.
c) To find the angular velocities of the two wheels during the constant velocity part of the trip, we can use the formula:
Angular velocity = Velocity / Radius
We know the velocity during the constant velocity phase and the radii of the wheels.
d) To find the angular accelerations of the two wheels during the first two minutes, we need to calculate the change in angular velocity and divide it by the time:
Angular acceleration = (Change in angular velocity) / Time
The change in angular velocity can be calculated by subtracting the initial angular velocity from the final angular velocity, which can be found using the formula for angular velocity.
To summarize:
a) Calculate distances traveled during each phase and add them together to find the total distance traveled.
b) Calculate angles using the formula Angle = Distance / Radius.
c) Calculate angular velocities using the formula Angular velocity = Velocity / Radius.
d) Calculate angular accelerations using the formulas Angular acceleration = (Change in angular velocity) / Time and Angular velocity = Velocity / Radius.