Polar Coordinates and Distance

Question

An air traffic controller‘s radar display uses polar coordinates. A passing plane is detected at 285° counterclockwise from north at a distance of 3 miles from the radar. 30 seconds later the plane is detected at 225° and 2 miles. Estimate the plane speed in miles per hour.

Answer 1

You need to convert each polar coordinate to rectangular coordinates, but first we need to re-write the angle measurements in terms we can utilize.  Due North on a compass is 90 degrees on the unit circle.  Therefore 285 from north would be 90 + 285 = 375 or 15 degrees.  225 would be 225 + 90 = 315.

 

Standard polar to rectangular coordinates: x = r cos (theta) and y = r sin (theta).

 

So the first point would be x =3 cos (15) and y = 3 sin (15) or x = 2.898 and y = .776

 

The second point would be x = 2 cos (315) and y =2 sin (315) or x = 1.414 and y = -1.414

 

Now find the distance between the points (distance formula):

 

sqrt[(2.898-1.414)² + (.776 - -1.414)²] = 2.645 miles

 

Now find the speed.  30 seconds = 30/[(60)(60)] hours = 1/120 hours

 

speed = distance/time = 2.645/(1/120)  = 317.4 mph