A 200 toy car is placed on a narrow 64--diameter track with wheel grooves that keep the car going in a circle. The 1.5 track is free to turn on a frictionless, vertical axis. The spokes have negligible mass. After the car's switch is turned on, it soon reaches a steady speed of 0.77 relative to the track.

To calculate the speed at which the toy car is traveling relative to an observer outside the track, we can use the concept of angular velocity. Angular velocity is the rate at which an object rotates around a central axis. In this case, the central axis is the vertical axis of the track.

First, we need to find the angular velocity of the toy car. The angular velocity can be calculated using the formula:

ω = v / r

where:
ω is the angular velocity,
v is the linear velocity (speed) of the car relative to the track, and
r is the radius of the circular track.

In this case, the linear velocity of the car relative to the track is given as 0.77 m/s, and the radius of the track is half of the diameter, which is 64 / 2 = 32 cm = 0.32 m.

Substituting these values into the formula, we have:

ω = 0.77 m/s / 0.32 m
ω ≈ 2.406 rad/s

The angular velocity of the toy car relative to the track is approximately 2.406 radians per second.

To find the speed of the toy car relative to an observer outside the track, we need to consider the relative motion of the observer and the track. Since the track is free to turn on a frictionless vertical axis, the observer and the track will have the same angular velocity.

Therefore, the speed of the toy car relative to the observer outside the track is also 2.406 radians per second.

Please note that the given information does not provide the distance traveled by the toy car. If you have any additional information, I can assist you further.