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

2. Vector vs. Scalar Review
All physical quantities encountered in this text will be either a scalar or a vector
A vector quantity has both magnitude (size) and direction
A scalar is completely specified by only a magnitude (size)

3. Vector Notation When handwritten, use an arrow:
When printed, will be in bold print with an arrow:
When dealing with just the magnitude of a vector in print, an italic letter will be used: A

4. Properties of Vectors Equality of Two Vectors
Two vectors are equal if they have the same magnitude and the same direction
Movement of vectors in a diagram
Any vector can be moved parallel to itself without being affected

5. More Properties of Vectors
Negative Vectors
Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)
Resultant Vector
The resultant vector is the sum of a given set of vectors

6. Adding Vectors
When adding vectors, their directions must be taken into account
Units must be the same
Algebraic Methods
More convenient

7. Notes about Vector Addition
Vectors obey the Commutative Law of Addition
The order in which the vectors are added doesn’t affect the result

8. Vector Subtraction Special case of vector addition
Add the negative of the subtracted vector
Continue with standard vector addition procedure

9. Multiplying or Dividing a Vector by a Scalar
The result of the multiplication or division is a vector
The magnitude of the vector is multiplied or divided by the scalar
If the scalar is positive, the direction of the result is the same as of the original vector
If the scalar is negative, the direction of the result is opposite that of the original vector

10. Components of a Vector A component is a part
It is useful to use rectangular components
These are the projections of the vector along the x- and y-axes

11. Example 1
Find the components of the following vector if it has a magnitude of 300m/s and the vector makes a 37o angle with the x-axis.

12. Example 2
Find the components of the following vector.
θ=400
500m

13. Components of a Vector, cont.
The x-component of a vector is the projection along the x-axis
The y-component of a vector is the projection along the y-axis
Then,

14. More About Components of a Vector
The previous equations are valid only if θ is measured with respect to the x-axis
The components can be positive or negative and will have the same units as the original vector

15. More About Components, cont.
The components are the legs of the right triangle whose hypotenuse is

16. Adding Vectors Algebraically
Choose a coordinate system and sketch the vectors
Find the x- and y-components of all the vectors
Add all the x-components
This gives Rx:

17. Adding Vectors Algebraically, cont.
Add all the y-components
This gives Ry:
Use the Pythagorean Theorem to find the magnitude of the resultant:
Use the inverse tangent function to find the direction of R:

18. Signs of Components 1st Quadrant x-component is positive
y-component is positive

19. Signs of Components 2nd Quadrant x-component is negative
y-component is positive

20. Signs of Components 3rd Quadrant x-component is negative
y-component is negative

21. Signs of Components 4th Quadrant x-component is positive
y-component is negative

22. Example 3
Each of the displacement vectors A and B shown in the figure has a magnitude of 3.00 m. Find the magnitude and the direction of 4A + 5B by components method.

23. Motion in Two Dimensions
Using + or – signs is not always sufficient to fully describe motion in more than one dimension
Vectors can be used to more fully describe motion
Still interested in displacement, velocity, and acceleration

24. Displacement
The position of an object is described by its position vector,
The displacement of the object is defined as the change in its position

25. Example 4
A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal. Next, it travels 135 ft at an angle of 40.0° below the horizontal.

26. Example 5
Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x-axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x-axis. Find (a) the vector sum A + B and (b) the vector difference A – B.

27. Example 6
A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

28. Example 7
An airplane starting from airport A flies 300 km east, then 350 km at 30.0° west of north, and then 150 km north to arrive finally at airport B. (a) The next day, another plane flies directly from A to B in a straight line. In what direction should the pilot travel in this direct flight? (b) How far will the pilot travel in the flight? Assume there is no wind during either flight.

29. Velocity
The average velocity is the ratio of the displacement to the time interval for the displacement
The instantaneous velocity is the limit of the average velocity as Δt approaches zero
The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion

30. Acceleration
The average acceleration is defined as the rate at which the velocity changes
The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero

31. Ways an Object Might Accelerate
The magnitude of the velocity (the speed) can change
The direction of the velocity can change
Even though the magnitude is constant
Both the magnitude and the direction can change

32. Projectile Motion
An object may move in both the x and y directions simultaneously
It moves in two dimensions
The form of two dimensional motion we will deal with is called projectile motion

33. Assumptions of Projectile Motion
We may ignore air friction
We may ignore the rotation of the earth
With these assumptions, an object in projectile motion will follow a parabolic path

34. Rules of Projectile Motion
The x- and y-directions of motion are completely independent of each other
The x-direction is uniform motion
ax = 0
The y-direction is free fall
ay = -g
The initial velocity can be broken down into its x- and y-components

35. Projectile Motion

36. Projectile Motion at Various Initial Angles
Complementary values of the initial angle result in the same range
The heights will be different
The maximum range occurs at a projection angle of 45o

37. Some Details About the Rules
x-direction
ax = 0
x = vxot
This is the only operative equation in the x-direction since there is uniform velocity in that direction

38. More Details About the Rules
y-direction
free fall problem
a = -g
take the positive direction as upward
uniformly accelerated motion, so the motion equations all hold

39. Velocity of the Projectile
The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point
Remember to be careful about the angle’s quadrant

40. Problem-Solving Strategy
Select a coordinate system and sketch the path of the projectile
Include initial and final positions, velocities, and accelerations
Resolve the initial velocity into x- and y-components
Treat the horizontal and vertical motions independently

41. Problem-Solving Strategy, cont
Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile
Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile

42. Some Variations of Projectile Motion
An object may be fired horizontally
The initial velocity is all in the x-direction
vo = vx and vy = 0
All the general rules of projectile motion apply

43. Non-Symmetrical Projectile Motion
Follow the general rules for projectile motion
Break the y-direction into parts
up and down
symmetrical back to initial height and then the rest of the height

44. Example 8
An artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. To clear an avalanche, it explodes on a mountainside 42.0 s after firing. What are the x- and y-coordinates of the shell where it explodes, relative to its firing point?

45. Example 9
A child kicks a rock off the top of a 30m cliff, if the rock lands 4.2m from the base of the cliff, what horizontal velocity was given to the rock?

46. Example 10
Tom the cat is chasing Jerry the mouse across the surface of a table 1.5 m above the floor. Jerry steps out of the way at the last second, and Tom slides off the edge of the table at a speed of 5.0 m/s. Where will Tom strike the floor, and what velocity components will he have just before he hits?

47. Example 11
An rescue plane drops a package of emergency rations to stranded hikers. The plane is traveling horizontally at 60.0 m/s at a height of 300 m above the ground. Assume the hikers are 200 meters away from where the package was released (a horizontal distance) and the package landed somewhere in front of them. How far do the hikers have to walk to pick up the package from the ground?

48. Example 12
A home run is hit in such a way that the baseball just clears a wall 21 m high, located 130 m from home plate. The ball is hit at an angle of 35° to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume that the ball is hit at a height of 1.0 m above the ground.)

49. Relative Velocity
Relative velocity is about relating the measurements of two different observers
It may be useful to use a moving frame of reference instead of a stationary one
It is important to specify the frame of reference, since the motion may be different in different frames of reference
There are no specific equations to learn to solve relative velocity problems

50. Example 13
A jet airliner moving initially at 300 mi/h due east enters a region where the wind is blowing at 100 mi/h in a direction 30.0° north of east. What is the new velocity of the aircraft relative to the ground?

51. Example 14
A river flows due east at 1.50 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 10.0 m/s due north relative to the water. (a) What is the velocity of the boat relative to the shore? (b) If the river is 300 m wide, how far downstream has the boat moved by the time it reaches the north shore?

52. Example 15
A science student is riding on a flatcar of a train traveling along a straight horizontal track at a constant speed of 10.0 m/s. The student throws a ball along a path that she judges to make an initial angle of 60.0° with the horizontal and to be in line with the track. The student’s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does the ball rise?