Help with this question please!?

Question

Two people run at the same average speed. Runner A cuts directly north across the diameter of the circular track, while Runner B takes the full semicircle to meet his partner on the opposite side of the track. Assume their common average speed is 2.70 m/s and the track has a diameter of 150 m. (a) How many seconds ahead of runner B does runner A arrive? (b) How do their travel distances compare? (c) How do their displacements compare? (d) How do their average velocities compare?

Does anyone know how to work this problem? Your help will be greatly appreciated!

Answer 1

First, let's look at translational comparisons.

Runner A travels 150m North at a speed of 2.7 m/s
so the time to arrive at the apex of the track is
150/2.7 seconds
or 55.6 seconds

Runner B travels half the circumference of a circle with diameter of 150 meters
the distance traveled is
150*pi/2
or 235 meters at 2.7 m/s
for a time of 87 seconds

Now look at velocity

Runner A has a constant velocity of 0,2.7 m/s (vector quantity)

Runner B, however, has a constant rotational velocity of
2.7/75 radians per second

The vector quantity is constantly changing depending on the angle of Runner B with respect to the compass. At the Southern starting point, the vector is 2.7, 0. At the Easterly tip, when the displacement angle is 90 degrees from the start, the vector is 0, 2.7, and at North, the vector is -2.7,0

The vector of Runner B is constantly changing. The equation of a circle with the center at the origin of a Cartesian coordinate system, is

R^2=x^2+y^2

If you look at the equation of a circle in polar coordinates, r=constant.

This question is the beginning of a look at why a rotating frame of reference is used to describe motion of spinning objects. In a rotational frame of reference, an object moving at constant speed along a constant radius from a fixed point has zero angular acceleration w/r/t the point. Whereas, if the motion is described in x,y coordinates, the object has variable velocity that is complex to manage.

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