1. Two-Dimensional Motion and Vectors
Chapter 3
Two-Dimensional Motion and Vectors

2. Introduction to Vectors
Section 3.1
Introduction to Vectors

3. Scalars and Vectors
Scalar: a physical quantity that has only a magnitude but no direction
Vector: a physical quantity that has both a magnitude and direction

4. Scalar or vector Are the following quantities scalars or vectors?
Weight
Displacement
Speed
Time
Velocity
Force

5. Scalars and Vectors
In this book, vectors are indicated by the use of boldface type
In this book, scalars are indicated by the use of italics

6. Vectors Vectors can be added graphically
They must describe similar quantities
They must have the same units
The answer found by adding two vectors is called the resultant (vector sum)

7. Triangle method of addition
Use a reasonable scale
Draw the tail of one vector starting at the tip of the other
Draw the resultant vector starting at the tail of the first vector to the tip of the last vector

8. Vectors continued Vectors can be added in any order
Commutative property of vectors

9. Vectors continued To subtract a vector, add the opposite
When you multiply or divide a vector by a scalar, the result is a vector
i.e. Twice as fast
Always REMEMBER to indicate direction
North of east
South of west

10. Section 3.2
Vector Operations

11. Coordinate Systems in 2 Dimensions
In Chapter 2 we looked at motion in 1 dimension

12. Coordinate Systems in Two Dimensions
Adding another axis helps us describe two dimensional motion
It also simplifies analysis of motion in one dimension
Fig 3.6
In a, if the direction of the plane changes, we must turn the axis again
It would also be difficult to describe another moving object if it is not traveling in a different direction
Having 2 axes simplifies this significantly

13. Coordinate Systems in Two Dimensions
There are no firm rules for applying coordinate systems
However, you must be consistent
It is usually best to use the system that makes solving the problem the easiest
This is why we use the coordinate system

14. Determining Resultant Magnitude and Direction
In Section 3.1 we did this graphically
This takes too long and is only accurate if you draw VERY CAREFULLY
A better way is by using the Pythagorean theorem

15. Determining Resultant Magnitude and Direction
In order to find the direction of the resultant, we can use the tangent function

16. Problem
A Plane travels from Houston, Texas, to Washington, D.C., which is 1540 km east and 1160 km north of Houston. What is the total displacement of the plane?

17. Resolving Vectors into Components
The horizontal and vertical parts that add up to give the resultant are called components.
The x component is parallel to the x-axis and the y component is parallel to the y-axis
Any vector can be described by a set of perpendicular components.
Fig 3.10

18. Resolving Vectors into Components
For a refresher on trigonometry, see Appendix A in the back of the book.

19. Problems
An arrow is shot from a bow at an angle of 25o above the horizontal with an initial speed of 45 m/s. Find the horizontal and vertical components of the arrow’s initial velocity.
The arrow strikes the target with a speed of 45 m/s at an angle of -25o with respect to the horizontal. Calculate the horizontal and vertical components of the arrow’s final velocity.

20. Adding vectors that are not perpendicular
A plane travels 50km at an angle of 35o to the ground, then climbs at only 10o to the ground for 220 km.
These vectors do not form a right triangle, so we can’t use the tangent function or the Pythagorean theorem when adding them
However, we can resolve each of the plane’s displacement vectors into their x and y components

21. Adding vectors that are not perpendicular
We can then use the Pythagorean theorem and tangent function on the vector sum of the two perpendicular components of the resultant

22. Adding Vectors Algebraically
A hiker walks 25.5 km from her base camp at 35o south of east. On the second day, she walks 41.0 km in a direction 65o north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

23. Adding vectors algebraically
Step 1: Select a coordinate system, draw a sketch of the vectors to be added and label each vector

24. Adding vectors Algebraically
Step 3: Find the x and y components of the total displacement.
Step 4: Use the Pythagorean theorem to find the magnitude of the resultant vector
Step 5: Use a suitable trigonometric function to find the angle the resultant vector makes with the x-axis
Step 6: Evaluate your answer.

25. HW Assignment Pg 91: Practice 3a - 1, 2 Pg 94: Practice 3b - 2, 4, 6
Pg 97: Practice 3c - 2, 3, 4

26. Section 3.3
Projectile Motion

27. Two-Dimensional Motion
The velocity, acceleration and displacement of an object thrown into the air don’t all point in the same direction.
We resolve vectors into components to make it simpler.
At the end, we can recombine the components to determine the resultant.

28. Components simplify motion
As a long jumper approaches his jump, he runs in a straight line.
When he jumps, his velocity has both horizontal and vertical components.
To analyze his motion we apply the kinematic equations to one direction at a time

29. Projectile Motion Free fall with an initial horizontal velocity
Projectiles follow parabolic trajectories
However, there is air resistance at the surface of the earth and horizontal velocity slows down.
In this class, we will consider the horizontal velocity to be constant.

30. Vertical motion of a projectile

31. Vertical motion of a projectile that falls from rest

32. Horizontal motion of a projectile

33. Problem
People in movies often jump from buildings into pools. If a person jumps from the 10th floor (30.0 m) to a pool that is 5.0 m away from the building, with what initial velocity must the person jump?

34. Objects launched at an angle
For an object launched at an angle, the sine and cosine functions can be used to find the horizontal and vertical components of the vi.
Use components to analyze objects launched at an angle

35. Problem
A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, the zookeeper kneels 10.0 m from the light pole, which is 5.00 m high. The tip of her gun is 1.00 m above the ground. The monkey tries to trick the zookeeper by dropping a banana, then continues to hold onto the light pole. At the moment the monkey releases the banana, the zookeeper shoots. If the tranquilizer dart travels at 50.0 m/s, will the dart hit the monkey, the banana, or neither one?

36. Problem
A golfer practices driving balls off a cliff and into water below. The cliff is 15 m from the water. If the golf ball is launched at 51 m/s at an angle of 15o, how far does the ball travel horizontally before hitting the water?