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CHAPTER 3
Two-Dimensional Motion and
Vectors
Representations:
x
y
(x, y)
(x, y) (r, !)
VECTOR quantities:
Vectors have magnitude and direction.
Other vectors: velocity, acceleration, momentum, force …
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)
Vector Components
Cartesian components are projections along the x- and y-axes Ax = A cos! Ay = A sin!
Going backwards,
A = Ax^2 + Ay^2 and! = tan"^1
Ay Ax
Example 3.1a
The magnitude of (A-B) is :
a) < b) = c) >
Example 3.1b
The x-component of (A-B) is:
a) < b) = c) >
Example 3.1c
The y-component of (A-B) > 0
a) < b) = c) >
Example 3.
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?
14.55 miles, at 20.10 degrees Alice
Bob
Carol
Arcsin, Arccos and Arctan: Watch out!
same sine
same cosine
same tangent
Arcsin, Arccos and Arctan functions can yield wrong angles if x or y are negative.
2-dim Motion: Velocity
Graphically,
v = "r / "t
It is a vector (rate of change of position)
Trajectory
Multiplying/Dividing Vectors by Scalars, e.g. " r / " t
- (^) 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
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)
Example 3.4a
h
D
v 0
The Y-component of v at A is : a) < b) 0 c) >
Example 3.4b
h
D
v 0
a) < b) 0 c) >
The Y-component of v at B is
Example 3.4c
h
D
v 0
a) < b) 0 c) >
The Y-component of v at C is:
Example 3.4d
h
D
v 0
a) A b) B c) C d) Equal at all points
The speed is greatest at:
Example 3.4e
h
D
v 0
a) A b) B c) C d) Equal at all points
The X-component of v is greatest at:
Example 3.4f
h
D
v 0
a) A b) B c) C d) Equal at all points
The magnitude of the acceleration is greatest at:
Range Formula
- Good for when yf = yi x = vi , xt
y = vi , yt!
gt^2 = 0
t =
2 vi , y g
x =
2 vi , xvi , y g
2 vi^2 cos " sin " g
x =
vi^2 g
sin 2 "
Range Formula
R = (^) • Maximum for !=45°
vi^2 g
sin 2!
Example 3.5a
100 m
A softball leaves a bat with an initial velocity of 31.33 m/s. What is the maximum distance one could expect the ball to travel?
Example 3.
299 m
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?
h
D
v 0
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?
B. Considering the effects of gravity, at what angle should the hunter aim the arrow relative to the ground?
Example 3.7, Shoot the Monkey
!=Arctan(h/L)=26.6°
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.
Must find v Solution: 0,y/vx in terms of h and L
- Height of arrow
- Height of monkey
- Require monkey and arrow to be at same place
Aim directly at Monkey!
yarrow = v 0 , yt!
gt^2
y monkey = h!
gt^2
h!
gt^2 = v 0 , yt!
gt^2
h = v 0 , yt = v 0 , y
L
vx
v 0 , y vx
h L
Example 3.
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
Example 3.11a Three airplanes, A, B and C, with identical air speeds fly from Williamston, MI, towards Tallahassee, FL, which is directly south. “A” flies on Monday when there is a strong wind from the west. “A” aims the plane south but is blown off course. “B” also leaves Monday, but aims a bit into the wind and lands in Tallahassee. “C”flies on Tuesday, a calm and windless day, and flies directly to Tallahassee. Which plane(s) has(have) the HIGHEST ground speed?
A) A B) B C) C D) A and B E) B and C
Example 3.11b
Three airplanes, A, B and C, with identical air speeds fly from Williamston, MI, towards Tallahassee, FL, which is directly south. “A” flies on Monday when there is a strong wind from the west. “A” aims the plane south but is blown off course. “B” also leaves Monday, but aims a bit into the wind and lands in Tallahassee. “C”flies on Tuesday, a calm and windless day, and flies directly to Tallahassee. Which plane(s) has(have) the LOWEST ground speed?
A) A B) B C) C D) A and B E) B and C
