Create a name for each point in order you are traveling. Then find the direction and magnitude vector for each path. Lastly, put them in component form of each path.

Question

CVG Airport: (-8,-7)

Kenwood Mall: (4,2)

Northgate Mall: (-6,4.5)

Riverbend Music Center: (2,-6)

Cincinnati Zoo: (-2,-1)

Answer 1

The plot of these points looks like this:

To calculate velocity, you would first need to state that the ordered pairs are given as distances in some unit of length, I'll say "miles". Then, secondly, you would need to specify a time. Velocity cannot be calculated without knowing the time because velocity is displacement/time. Using the coordinate points will allow you to calculate displacement but you need time to get velocity. I might have written the problem like this:

Starting at the airport, create an optimal path to travel to each point. Assume that you were able to travel in straight lines, at the same speed, between the points and that you stopped for 1 hour at each mall, 30 minutes at the music center and 2 1/2 hours at the zoo and that the overall time from start to arrival at the final location was 5 hours. Calculate the velocity of each section. Also, break down the velocity into its components for each section. Finally, calculate the overall velocity of the trip.

To solve this, you would first determine the displacement traveled in each section and add them up to give you the total distance traveled. I would travel from the airport (C) to R to Z to K to N. The distance between C-R is sqrt(6² + 1²) = 6.0828. The distance between R-Z is 5. The distance between Z-K is sqrt(6² + 3²) = 6.7082. The distance between K-N is sqrt(10² + 2.5²) = 10.3078 so the total distance is 6.0828 + 5.0000 + 6.7082 + 10.3078 = 28.099. The time actually driving would be 5 hours but I would need to subtract the time I spent at each place. I spent 0.5 hr at R (Riverbend Music Center), 2.5 hours at the Z (Cincinnati Zoo), and 1 hour at K (Kenwood Mall) but the time spent at N (Northgate Mall) doesn't matter because the overall time is from the start to the arrival at the last place. So, the "non-driving time is 0.5 + 2.5 + 1 = 4 hours making my driving time 5 - 4 = 1 hr. So my speed was 28.099 miles/1 hr = 28.1 mi/hr. That will also be the magnitude of the velocity for each segment. The direction is calculated by taking the inverse tangent of each x and y component. Example, the direction C-R is tan^-1(1/6) = 9.5° north of east. The velocity components can be calculated by using the speed times the cosine of the angle and the speed times the sine of the angle. Example, the velocity in the x-direction for C-R is 28.099cos(9.5°) = 27.7 mi/hr east and the velocity in the y-direction is 28.099sin(9.5°) = 4.6 mi/hr north.

The overall velocity is the total displacement divided by the total time. The total displacement is the direct path between C and N which is sqrt(2² + 11.5²) = 11.6726 mi and the total time is 5 hours so the magnitude of the velocity is 11.6726/5 = 2.3 mi/hr. The direction is tan^-1(11.5/2) = 80.1° north of east