Resource Lesson
Vector Resultants: Average Velocity
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A Graphical Approach: Head-To-Tail
We will first show you a graphical method, called "head-to-tail, to add together two concurrent vectors.
The terms
"head" and "tail"
are labeled as shown:
Concurrent
vectors share a common starting point.
As you view the animation shown below, note that the "tail" or start of the first vector is initially placed at the origin of the co-ordinate system. At its "head" you initially place a second "imaginary" co-ordinate, or reference, system, and then the "tail" of the second vector. If a third vector were to be added, another imaginary co-ordinate system would be placed at the head of the second vector and the tail of the third vector would begin there.
The green vector represents the sum of the two vectors, or the
resultant
. It begins at the origin of the original co-ordinate both ends and points towards the head of the last vector being added. The length of the resultant is called it
magnitude
, the angle that the resultant makes with the original x-axis is called its
direction
.
You would use the Pythagorean Theorem to mathematically calculate the resultant's magnitude. To determine its directional angle, θ, you would use the trig function tangent.
Let's use this method to answer some questions about a child's journey between two houses in which he ran 10 meters North and then 10 meters West.
We can readily tell that the distance he traveled was 20 meters, but what was his displacement? Was it equal to 20 meters, less than 20 meters, or greater than 20 meters?
To determine his actual displacement we need to draw a "head-to-tail" diagram of his trip and calculate his resultant.
The magnitude of his resultant vector equals the length of the hypotenuse in the diagram shown above.
As you can see, his displacement was less than the actual distance he traveled. It was 14.1 meters in a direction of 45º W of N. We know the angle must equal 45º since we are working in an isosceles right triangle.
Refer to the following information for the next six questions.
Now suppose our child took 4 seconds to get to his destination.
1. What was his average speed?
2. What was the magnitude of his average velocity?
3. Why are your answers to #1 and #2 not the same?
4. What is the direction of his average velocity?
5. If after an additional 4 seconds the child manages to retrace his steps and return to his original starting point, what total distance did he travel?
6. What was his final net displacement for the entire 8 seconds?
An Analytical Approach: x|y Charts
But what if his path had been more convoluted? How would we determine his resultant if it involved more than two "sections"?
When more than two vectors are added together, it is often more convenient to make use of an
x|y chart
as well as a graphical display.
Let's augment our original story. After the child walked 10 meters North and then 10 meters West, he decides to go 25 meters South, 8 meters further West, 7 meters back North and finally 14 meters East. Graphically his path would looks like that shown below.
When organized in an x|y chart this information would look like:
x
y
-10 m
10 m
-8 m
-25 m
14 m
7 m
- 4 meters
- 8 meters
This confirms that his final displacement has a magnitude of 8.9 meters at 243.3º.
Refer to the following information for the next four questions.
Now suppose our child took 5 minutes to cover the path outlined above and reach his final destination.
1. What was his average speed?
2. What was the magnitude of his average velocity?
3. Why are your answers to #1 and #2 not the same?
4. In what direction would he need to walk to return home?
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