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INTRODUCTORY PHYSICS HUNTER COLLEGE

VECTORSPurpose of Exercise:

Students will be able to calculate the resultant vectors’ magnitude and direction by

● Resolving vectors into their components

● Using the rules of vector addition

● Graphing the vectors on a cartesian coordinate system

● Using basic trigonometry to determine the angle of the resultant vector

Background Information

● Vector quantities are any quantity that has a size (magnitude) and direction. Such quantities are

displacement, velocity, acceleration, and force.

● A scalar quantity is any quantity with a magnitude only. Velocity without its direction would be

the scalar quantity we call speed. Distance, time, temperature, and mass are examples of scalar quantities.

Graphically Adding Components

There are graphical methods to add vectors. The tip to tail and parallelogram methods both are visual aids to adding vectors. With proper scaling and a protractor, you can measure the magnitude of the resultant vector with a ruler and determine its angle with a protractor. Both methods are displayed in the figure below:

https://reepingdavid.weebly.com/uploads/2/7/2/3/27230509/11299_orig.png

Adding Vectors Using Components

Any vector can be expressed as the sum of two other vectors, which are called its components. This is useful and a more practical means of adding vectors. Determining the components of a vector is to resolve a vector into its components. The figure shows vector V resolved (left) into its components Vx

and Vy (right)

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Resolving vectors into its components is useful when adding vectors that are not one dimensional and are in different directions. For example, you are walking North 4 miles per hour while the wind is blowing 3 miles per hour. You are blown off course and your resultant velocity is 5 mph at 53.1⁰, above the horizontal axis, or 53.1⁰North of East.

Steps to adding vectors using components:

1. Draw a diagram; add the vectors graphically.

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using sines and cosines.

5. Add the components in each direction.

6. To find the length (or magnitude of the vector) use the Phythagorean Theorem, or distance

formula,

7. To find the direction of the vector use, use:

Pre-Lab Questions

1. Resolve a vector A of the 5.0 unit at 53.1⁰ into its components.

Ax = Ay =

2. Vector B has components, Bx = -3.0 units and By = +4.0 units, calculate the magnitude and direction of B.

3. Although vector A has the same magnitude as vector B, why are they not the same vector?

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Procedures: Vector Addition

Part I: Explore 2D

1. Open the vector simulation: https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_en.html

2. Open Explore 2D by clicking on the icon above. Your screen will open the simulation like the one in the figure below.

4. Re-orient the origin on the graph. Go to the origin (0,0). Click and drag it to the center of the

graph: Your graph should look like this .

5. Add vector a by clicking and dragging vector a from the lower right panel on your screen . 6. Place the tail of a at the origin.

7. Put your curser on the tip of the vector arrow; the hand cursor will appear. 8. Click and drag the tip of the vector to change its magnitude and direction.9. Set a to have a magnitude of 9 units and direction of 0⁰, so that its coordinates are (9,0).

10. Now add vector b to the graph. Put the tail at the zero. Set its magnitude to 12 units and its direction at 90 ⁰, so that its coordinates are (0,12).

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11. Use the Pythagorean Theorem to determine the magnitude and the angle, relative to the x-axis.

12. If you calculated correctly, you should have 15 units at 53.1⁰ (above the positive x-axis). Confirm

this by checking the Sum and Values options in the upper-right corner on your screen .

13. Record your vector measurements and calculation in Table 1 below. Explore the various ways you can read vector values in the simulation. Checking the values option in the upper right corner is one way. Another way is to click a vector and read off the values in the grey rectangle above the graph.

14. Now, uncheck the Sum option.

15. Click on the vectors a and b to drag and move them around the coordinate system (graph). Be careful to only click and drag the body of the arrow, not the tip. Clicking the tip will change the vector value.

16. Table one has specified values for you to assign vectors a and b. For each trial fill in the table with the information requested in the top row of the table. In other words, complete the table by “resolving” the vectors into their components then sum to get the resultant s vector’s magnitude and determine its direction. Describe the quadrant where s is located according to its components and calculated angle. The first entry has been completed for you, as a guide.

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Table 1 Vector Label

Vector Magnitude [units U]

Vector Direction

ϴv

Horizontal Component

[units U]

Vertical Component

[units U]

Resultant Vector S

Magnitude

[units U]

S = √( ax 2+ bx

2 )

Resultant

Vector Direction

ϴs = tan-1(sy / sx )

In what quadrant is the Resultant S?

Trial 1 a 9 0⁰ ax= 9 ay= 0 10 53.1⁰ I

b 12 90⁰ bx= 0 by= 12

s Calculate -> sx = ax+ bx

sx = 9

sy = ay+ by

sy = 12

Trial 2 a 9 0⁰ ax= ay=

b 12 270⁰ bx= by=


sx =

sy = ay+ by

sy =


b 12 270⁰ bx= by=


sx =

sy = ay+ by

sy =


b 12 90⁰ bx= by=


sx =

sy = ay+ by

sy =

Question 1: How did the magnitudes and direction of s change as you varied vectors a and b?

Part II: Lab

1. Click on the home tab at the bottom center of the screen.

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2. Click on the Lab tab, , to open the lab simulation.

3. On the upper right panel check the value and angle options .4. Add a blue vector onto the graph by clicking and dragging a blue vector from the right lower

panel .

5. Put your curser on the tip of the vector arrow; the hand cursor will appear. 6. Click and drag the tip of the vector to change its magnitude and direction.7. Choose a magnitude of the vector between 8.0 and 12.0 units and its direction between 10.0⁰

and 80.0⁰.8. Add 2 more blue vectors on the graph. 9. Choose a magnitude between 0 and 12 units and a direction between 100⁰ and 170⁰, for one of

the vectors and a magnitude between 0 and 10 units with a direction between -10⁰ and -80⁰ for the other vector. Note: The simulation reads degrees from 0⁰ to 180⁰ counter-clockwise and 0⁰ to -180⁰ clockwise.

10. Click on the Components option to the right of the screen ; choose any of the three components options you like.

11. Now, observe the components of the three vectors and make a prediction of what the resultant sum of these 3 vectors will be. Estimate the magnitude and direction, without making any calculations. You may want to line up your 3 vectors to help you visualize the resultant like in the figure below. To move the vectors, click on the body of the arrow then drag around the graph to its desired location.

12. On the same graph, create a vector that best estimates your prediction by adding an orange

vector and that shows your prediction. 13. Take a screen shot. Include the 3 blue vectors and your prediction. It should look like this

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14. Now select the sum option for the blue vector . You will see the calculated resultant sum on your screen, the darker blue arrow. Briefly state how close was your prediction to the simulations resultant sum? Were the magnitude and direction of your prediction like the simulation’s resultant?

15. Unselect the sum, values, angle, component options, and remove your prediction vector (click and drag it back to the vector panel).

16. Click on the origin, (0,0), at the lower left of the screen and drag it to the center of the graph so you have the 4 quadrants of the coordinate system on the graph.

17. Now, place your 3 vectors on the graph so that the tails are at the origin like in the figure below.

18. Now, label your vectors a, b, c (in any order you choose) to fill in the table.

Vector Magnitude [Units] Direction θ Calculated

x-component

Calculated

y-component

a

b

c

19. Label the resultant sum S.20. Use your data from the table you just created to calculate the x-component and y-component of

S. Show your work:

Sx = Sy =

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21. Calculate the magnitude of S, show your calculation:

|S| =

22. Calculate the direction of S. Show your calculation:

ΘS =

23. Calculate the percent error between your calculation against the simulation’s calculation. You can read off the sum values, by selection the sum, values, and angle options on the upper right of the simulation screen.

24. Line the 3 blue vectors tip-to-tail, add an orange vector that represents the calculated sum, and take a screen shot of your result:

Part III: Vector Equations

1. Click on the home tab at the bottom center of the screen.

2. Click on the Lab tab, , to open the lab simulation.3. Create two vectors, a = 6 units at 0⁰ and vector b = 7 units at -90⁰, by clicking the base vector tab

in on the lower right of the screen and manually entering the component values of a and b.

4. Click the a + b = c equation at the top left tab above the graph.5. Record the values of vectors and record them in the table below.6. Repeat step 5 for the other two equation options.

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7. Calculate c for each equation. Show your work.8. Record your calculated values in the table below.9. Now, check the sum option on the simulation.10. Compare your results with the simulation’s calculation for each value of c determined from the

3 equations.

Table 3: Vector Equations

Equation x – component y-component calculated

Magnitude of c

Calculated

Direction of c θc

c = a + b cx = ax + bx = cy = ay + by =

c = a - b cx = ax - bx = cy = ay - by =

c + a + b = 0 cx =(ax + bx ) = cy = (ay + by)=

11. Show your calculations for each calculation of c, using the equations in Table 3 and compare your results with the simulation values for c.

12. How would vector c change if you multiplied vector a and b by 2? Change the magnitude and direction of vector c for:a. 2a + 2b, c = ?b. 2a – 2b, c = ?c. 2a + 2b + 2c = 0, c = ?

13. In your own words how would you describe what happens to a vector when you multiply it by a scalar?

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Post Lab Questions

1. You are in the middle of a large field. You walk in a straight line for 100 m, then turn left and walk 100 m more in a straight line before stopping. When you stop, you are 100 m from your starting point. By how many degrees did you turn? (a) 90°. (b) 120°. (c) 30°. (d) 180°. (e) This is impossible. You cannot walk 200 m and be only 100 m away from where you started.

2.

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