Chapter3 Nonlinear Motion
Vectors and the Parallelogram Rule

3. When vectors A and B are at an angle to each other, they add to produce the resultant C by the parallelogram rule. Note that C is the diagonal of a parallelogram where A and B are adjacent sides. Resultant C is shown in the first two diagrams, a and b. Construct the resultant C in diagrams c and d. Note that in diagram d you form a rectangle (a special case of a paraUelogram).

4. Below we see a top view of an airplane being blown offcourse by wind in various directions. Use the parallelogram rule to show the resulting speed and direction of travel for each case.

In which case does the airplane travel fastest across the ground?Slowest?

5. To the right we see top views of 3 motorboats crossing a river. All have the same speed relative to the water, and all experience the same water flow.

Construct resultant vectors showing the speed and direction of the boats.

a. Which boat takes the shortest path to the opposite shore?

b. Which boat reaches the opposite shore first?

c. Which boat provides the fastest ride?

6.

3. She tosses the ball along the dashed path. The velocity vector, complete with its horizontal and vertical components, is shown at position A. Careful]y sketch the appropriate velocity vectors with appropriate components for positions B and C.

a. Since there is no acceleration in the horizontal direction, how does the horizontal component of velocity compare for positions A, B, and C?

b. What is the value of the vertical component of velocity at position B?

c. How does the vertical component of velocity at position C compare with that of position A? 7. Prescriptive Analysis worksheet for Vectors

problem: Find magnitude & direction of the sum or difference of 2 vector given in Polar Form.

Prescriptive Analysis

1. Determine the reference system.

2. Draw diagrams of the 2 vectors (let's call them A and B, for now), with tails at origin.

3. Draw the sum (or difference) graphically using parallelogram method by completing the parallelogram and drawing the appropriate diagonal.

 

4. Move this resultant vector so that its tail is at the origin to determine correct quadrant of the resultant vector.

5. Use vector decomposition methods to obtain the x and y components of A and B .

6. Perform vector addition of x components and then of the y components.

7. Use Pythagoras's theorem to find the magnitude R of the resultant vector .

8. Use the inverse Tangent to obtain direction angle Å of R.

Conceptual Exercises

1. Given Ax and Å , draw A, find Ay and A.

2. Given (A,Å) and (B,Å), draw A, B, and A + B.

3. Given (A,ÅA), Rx and R, find (B,ÅB)

Examples.

1. a) Given: Ax = 5 m, Å = 45o, draw , find Ay and A.
b) Given Ay = 3 m and Å = 210o, draw A , find Ax and A.

2. A = (5 m, 60o), B = (3 m, 120o), draw A, B, and A + B.

3. Given A = (5,60o), Rx = 11.16, R = 14.5, where R = A + B, find (B, ÅB) 8. Ranking Task For Adding Displacement Vectors

Each of the figures below represent two consecutive displacements from the origin. The lengths of these displacement arrows are indicated by the numbers next to each arrow. Rank these situations on the basis of greatest total displacement from the origin. If you believe two or more situations represent the same net displacement from the origin, give them the same rank.

Greatest 1_______ 2_______ 3________ 4________ 5_______ 6_______ Least

The one with the greatest net displacement from the origin._________

Please carefully explain your reasoning

How sure are you of your ranking? Circle one:

Basically Guessed Sure Very Sure

1 2 3 4 5 6 7 8 9 10