In the design of a passenger bus, it is proposed to derive the motive power from the energy stored in a flywheel. The flywheel, which has a moment of inertia of 4.0 x 10^2 kgm^2, is accelerated to its maximum rate of rotation 3.0 x 10^3 revolutions per minute by electric motors at stations along the bus route. (i) calculate the maximum kinetic energy which can be stored in the flywheel. (ii) if, at an average speed of 36 kilometres per hour, the power required by the bus is 20 kW, what will be the maximum possible distance between stations on the level?

To answer the first part of the question and calculate the maximum kinetic energy stored in the flywheel, we can use the formula:

K.E. = (1/2) * I * w^2

Where:
K.E. = Kinetic energy
I = Moment of inertia of the flywheel
w = Angular velocity in radians per second

First, we need to convert the given angular velocity of 3.0 x 10^3 revolutions per minute into radians per second. There are 2π radians in one revolution, and 60 seconds in a minute. So, we can convert the angular velocity as follows:

w (in radians/second) = (3.0 x 10^3 rev/min) * (2π rad/rev) * (1 min/60 sec)

Simplifying the equation yields:
w ≈ 314.16 rad/sec

Now, we can substitute the values in the formula to calculate the maximum kinetic energy:

K.E. = (1/2) * (4.0 x 10^2 kgm^2) * (314.16 rad/sec)^2

Simplifying further:
K.E. ≈ 124.788 × 10^4 J

Therefore, the maximum kinetic energy stored in the flywheel is approximately 124.788 × 10^4 Joules (or 1.24 × 10^6 J).

Now, moving on to the second part of the question where we have to determine the maximum possible distance between stations on the level. We are given the average speed of the bus as 36 kilometers per hour and the power required by the bus as 20 kW.

Power (P) is defined as:

P = Work/Time

Since the work done by the bus is equal to the change in kinetic energy, we can rewrite the equation as:

P = ΔK.E./Time

We can rearrange the equation to solve for Time:

Time = ΔK.E./P

Substituting the given values, we get:

Time = (124.788 × 10^4 J)/(20,000 W)

Simplifying further:
Time ≈ 6.24 seconds

Now, since we know that Speed = Distance/Time, we can rearrange the equation to solve for Distance:

Distance = Speed * Time

Converting the average speed from kilometers per hour to meters per second:
Speed = (36 km/hr) * (1000 m/km) * (1 hr/3600 sec)

Simplifying further:
Speed ≈ 10 m/sec

Now, substituting the values into the Distance equation:

Distance = (10 m/sec) * (6.24 sec)

Simplifying further:
Distance ≈ 62.4 meters

Therefore, the maximum possible distance between stations on the level is approximately 62.4 meters.