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

2. Vectors have magnitude and direction.
VECTOR quantities:
Vectors have magnitude and direction. x y
(x, y)
Representations:
(x, y)
(r, q)
Other vectors: velocity, acceleration, momentum, force …

3. Vector Addition/Subtraction
2nd vector begins at end of first vector
Order doesn’t matter
Vector addition
Vector subtraction
A – B can be interpreted as A+(-B)

4. Cartesian components are projections along the x- and y-axes
Vector Components
Cartesian components are projections along the x- and y-axes
Going backwards,

5. Example 3.1a
The magnitude of (A-B) is :
a) <0
b) =0
c) >0

6. Example 3.1b
The x-component of (A-B) is:
a) <0
b) =0
c) >0

7. Example 3.1c
The y-component of (A-B) > 0
a) <0
b) =0
c) >0

8. Example 3.2
Alice and Bob carry a bottle of wine to a picnic site. Alice carries the bottle 5 miles due east, and Bob carries the bottle another 10 miles traveling 30 degrees north of east. Carol, who is bringing the glasses, takes a short cut and goes directly to the picnic site. How far did Carol walk?
What was Carol’s direction?
Carol
Bob
Alice
14.55 miles, at degrees

9. Arcsin, Arccos and Arctan: Watch out!
same cosine
same tangent
same sine
Arcsin, Arccos and Arctan functions can yield wrong angles if x or y are negative.

10. 2-dim Motion: Velocity v = Dr / Dt
It is a vector (rate of change of position)
Trajectory
Graphically,

11. Multiplying/Dividing Vectors by Scalars, e.g. Dr/Dt
Vector multiplied/divided by scalar is a vector
Magnitude of new vector is magnitude of orginal vector multiplied/divided by |scalar|
Direction of new vector same as original vector

12. Principles of 2-d Motion
X- and Y-motion are independent
Two separate 1-d problems
To get trajectory (y vs. x)
Solve for x(t) and y(t)
Invert one Eq. to get t(x)
Insert t(x) into y(t) to get y(x)

13. Projectile Motion X-motion is at constant velocity ax=0, vx=constant
Y-motion is at constant acceleration ay=-g
Note: we have ignored
air resistance
rotation of earth (Coriolis force)

14. Projectile Motion
Acceleration is constant

15. Pop and Drop Demo

16. The Ballistic Cart Demo

17. Finding Trajectory, y(x)
1. Write down x(t)
2. Write down y(t)
3. Invert x(t) to find t(x)
4. Insert t(x) into y(t) to get y(x)
Trajectory is parabolic

18. Example 3.3 v0
An airplane drops food to two starving hunters. The plane is flying at an altitude of 100 m and with a velocity of 40.0 m/s.
How far ahead of the hunters should the plane release the food? h X
181 m

19. Example 3.4a v0 h q D The Y-component of v at A is : a) <0 b) 0

20. Example 3.4b v0 h q D The Y-component of v at B is a) <0 b) 0

21. Example 3.4c v0 h q D The Y-component of v at C is: a) <0 b) 0

22. Example 3.4d v0 h q D The speed is greatest at: a) A b) B c) C
d) Equal at all points

23. Example 3.4e v0 h q D The X-component of v is greatest at: a) A b) B
c) C
d) Equal at all points

24. h D q v0
Example 3.4f
The magnitude of the acceleration is greatest at:
a) A
b) B
c) C
d) Equal at all points

25. Range Formula
Good for when yf = yi

26. Range Formula
Maximum for q=45

27. Example 3.5a
A softball leaves a bat with an initial velocity of m/s. What is the maximum distance one could expect the ball to travel?
100 m

28. h D q v0
Example 3.6
A cannon hurls a projectile which hits a target located on a cliff D=500 m away in the horizontal direction. The cannon is pointed 50 degrees above the horizontal and the muzzle velocity is 100 m/s. Find the height h of the cliff?
299 m

29. Example 3.7, Shoot the Monkey
A hunter is a distance L = 40 m from a tree in which a monkey is perched a height h=20 m above the hunter. The hunter shoots an arrow at the monkey. However, this is a smart monkey who lets go of the branch the instant he sees the hunter release the arrow. The initial velocity of the arrow is v = 50 m/s.
A. If the arrow traveled with infinite speed on a straight line trajectory, at what angle should the hunter aim the arrow relative to the ground?
q=Arctan(h/L)=26.6
B. Considering the effects of gravity, at what angle should the hunter aim the arrow relative to the ground?

30. Solution: Must find v0,y/vx in terms of h and L 1. Height of arrow
2. Height of monkey
3. Require monkey and arrow to be at same place
Aim directly at Monkey!

31. Shoot the Monkey Demo

32. Relative velocity
Velocity always defined relative to reference frame. All velocities are relative
Relative velocities are calculated by vector addition/subtraction.
Acceleration is independent of reference frame
For high v ~c, rules are more complicated (Einstein)

33. Example 3.8 1.067 hours = 1 hr. and 4 minutes 187.4 mph
A plane that is capable of traveling 200 m.p.h. flies 100 miles into a 50 m.p.h. wind, then flies back with a 50 m.p.h. tail wind.
How long does the trip take?
What is the average speed of the plane for the trip?
1.067 hours = 1 hr. and 4 minutes
187.4 mph

34. Relative velocity in 2-d
Sum velocities as vectors
velocity relative to ground = velocity relative to medium + velocity of medium.
vbe = vbr + vre
river wrt earth
Boat wrt earth
boat wrt river

35. pointed perpendicular to stream travels perpendicular to stream
2 Cases
pointed perpendicular to stream
travels perpendicular to stream

36. What is the resulting ground speed?
Example 3.9
An airplane is capable of moving 200 mph in still air. The plane points directly east, but a 50 mph wind from the north distorts his course.
What is the resulting ground speed?
What direction does the plane fly relative to the ground?
206.2 mph
14.0 deg. south of east

37. What is the plane’s resulting ground speed?
Example 3.10
An airplane is capable of moving 200 mph in still air. A wind blows directly from the North at 50 mph. The airplane accounts for the wind (by pointing the plane somewhat into the wind) and flies directly east relative to the ground.
What is the plane’s resulting ground speed?
In what direction is the nose of the plane pointed?
193.6 mph
14.5 deg. north of east