Adding Multiple Vectors
When we are adding a large amount of vectors (a complex vector motion) we can break down each vector into components and then add the components:
A radar outpost in Hawaii detects a UFO, it can turn on a dime and it’s motion is sporadic. The outpost gathered a series of veclocities and times which you have to add together and find out the UFO’s resultant displacement. The displacements began being recorded once the UFO was above the outpost, so the motion can be related to the outpost.
a) Find the resultant displacement, use components.
b) What does the resultant displacement tell us?
c) Does it matter in which order the velocities are added?
d) What is the average velocity? What does it mean?
Velocities:
320km/h [N76°E] for 30 min
340km/h [E35°S] for 20 min
100km/h [N40°W] for 15 min
20km/h [S30°W] for 1 hour
a)
Lets find the displacements, multiply the velocities by the amount of hours during which the UFO travelled at those velocities. (15 min = 1/4 hours)
160km [N76°E]
113km [E35°S]
25km [N40°W]
20km [S30°W]
Break down the displacements into components and add:
= 160 Ч Cos 76° [N] + 160 Ч Sin 76° [E] +
113 Ч Cos 35° [E] + 113 Ч Sin 35° [S] +
25 Ч Cos 40° [N] + 25 Ч Sin 40° [W] +
20 Ч Cos 30° [S] + 20 Ч Sin 30° [W]
= 38.7 [N] + 155.2 [E] + 92.6 [E] + 64.8 [S] + 19.2 [N] + 16.1 [W] + 17.3 [S] + 10.0 [W]
Remember that movement west is just a negative movement east, and that a movement north is a backwards movement south. We should get all horizontal motion in terms of one direction, and likewise for vertical motion. Lets write all vertical motion in terms of north and all horizontal motion in terms of east.
= 38.7 [N] + 155.2 [E] + 92.6 [E] – 64.8 [N] + 19.2 [N] - 16.1 [E] – 17.3 [N] – 10.0 [E]
= -24.2 [N] + 221.7[E]
= 24.2 [S] + 221.7[E]
Now that we have the two components of the resultant displacement we can use the pythagorean theorem and trigonometry to find the magnitude of and its direction.
Since the components form a 90° angle betweeen them:
| 
|І = 24.2І + 221.7І
| | = 223 km
Let Я be the angle between and the east direction
Tan Я = 24.2 [S] / 221.7[E]
Tan Я = 24.2 / 221.7
Tan Я = 0.10915
Я = 6.229
Therefore the UFO’s resultant displacement is 223km [E6°S] from the radar outpost. We first calculated how far is it, and then we calculated the angle of the resultant displacement away from its east component and toward its south component.
b) What does the displacement tell us? Doing all of this work for nothing is not very rewarding, you should always know what you are doing and why you are doing it. We have just added a series of motions together, they naturally occured head-to-tail, sequentially. The resultant displacement is the final spot at which the UFO was sighted, we can now dispatch a research team to that location. Remember that the resultant displacement is relative to the radar outpost!
Also, if the velocities began being recorded before/after the UFO was above the outpost we would not know to what initial position, on the ground, we should relate displacements to. This may sound confusing at first but by understanding reference points this will become clear.
c) Does the order of addition matter?
No. When adding vectors the order of addition does not matter. Vectors are concerned with “where you end up” as opposed to “how you got there” and every arrangement of the same set of motions will finish in the same place.
d) What is the average velocity? What does it mean?
Velocity is the change in displacement over time. The change in position was 223km [E6°S], the total time was
30min + 20min + 15min + 60min = 125min = 2.08hours.
Average velocity = 223km [E6°S] / 2.083hours
Average velocity = 107km/h [E6°S]
The average velocity means that if the UFO went in a straight line from the outpost to its final position, it would have to travel at 107km/h [E6°S] to arrive there after the same time it took for the sporadic motion. This is not very useful for this question but it might be for other types.