CHAPTER 3
Two-Dimensional Motion and
Vectors
VECTOR quantities:
Vectors have magnitude
and direction.
(x, y)
Representations:
(x, y)
y
(r, q)
x
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 q
Ay  A sin q
Going backwards,
A A  A
2
x
2
y
and
q  tan
1
Ay
Ax
Example 3.1
True or False:
a) The magnitude of A-B = 0
b) The x-component of A-B > 0
c) The y-component of A-B > 0
a) F
b) F
c) T
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 10 miles
traveling 30 degrees north of east. Carol, who
is bringing the glasses, takes a short cut which
goes directly to the picnic site.
How far did Carol walk?
What was Carol’s direction?
Carol
14.55 miles, at 20.10 degrees Alice
Bob
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
v = Dr / Dt
It is a vector
(rate of change of position)
Trajectory
Graphically,
Multiplying/Dividing Vectors by
Scalars, e.g. Dr / Dt
Vector multiplied/divided by scalar is a
vector
 Magnitude of new vector is magnitudeo the
orginial vector multiplied/divided by the
|scalar|
 Direction of new vector is the same or
opposite to original vector

Principles of 2-d Motion
•
•
•
X- and Y-motion are independent
Can be treated as two separate 1-d problems
To get trajectory (x vs. y)
1. Solve for x(t) and y(t)
2. Invert one Eq. to get t(x)
3. Insert t(x) into y(t) to get y(x)
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)
Projectile Motion
Acceleration
is constant
Pop and Drop Demo
The Ballistic Cart
Demo
1. Write down x(t)
x  v0, xt
Finding Trajectory, y(x)
2. Write down y(t)
1 2
y  v0, y t  gt
2
3. Invert x(t) to find t(x)
t  x / v0, x
4. Insert t(x) into y(t) to get y(x)
v0, y
1 g 2
y
x 2 x
v0, x
2 v0, x
Trajectory is parabolic
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?
181 m
h
X
Example 3.4
v0
h
q
1.
2.
3.
4.
5.
The
The
The
The
The
D
Y-component of v at A is (<0, 0, >0)
Y-component of v at B is (<0, 0, >0)
Y-component of v at C is (<0, 0, >0)
total velocity is greatest at (A,B,C)
X-component of v is greatest at (A,B,C)
1.>0
2. 0
3. <0
4. A
5. Equal at all points
Range Formula
 Good for when yf = yi
x  vi , xt
1 2
y  vi , y t  gt  0
2
2vi , y
t
g
2vi , x vi , y 2vi2 cos q sin q
x

g
g
2
vi
x  sin 2q
g
Range Formula
R
2
vi
g
sin 2q
 Maximum for q=45
Example 3.5a
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?
100 m
Example 3.5b
v0
h
q
D
A cannon aims a projectile at a target located on a cliff
500 m away in the horizontal direction and 75 meters
above the cannon. The cannon is pointed 50 degrees to
the horizontal. What muzzle velocity should the cannon
employ to hit the target?
75.4 m/s
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)=25.6
B. Considering the effects of gravity, at what
angle should the hunter aim the arrow relative to
the ground?
Must find v0,y/vx in terms of h and L
Solution:
1. Write height of arrow when x=L
2
v0, y
1 L
yarrow 
L g 2
vx
2 vx
2. Require y to be position of monkey at t=L/vx
2
1 2
1 L
ymonkey  h  gt  h  g 2
2
2 vx
3. Require monkey and arrow to be at same place
2
2
v
1 L
1 L
0, y
h g 2 
L g 2
2 vx vx
2 vx
v0, y h
Aim directly at Monkey!

vx
L
Shoot the Monkey Demo
Relative velocity

Velocity always defined relative to reference frame.

Relative velocities are calculated by vector
addition/subtraction.
Acceleration is independent of reference frame
For high, v ~c, rules are more complicated (Einstein)


All velocities are relative
Example 3.8
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
Relative velocity in 2-d
Sum velocities as vectors
 velocity relative to ground
= velocity relative to
medium + velocity of
medium.

vbe = vbr + vre
Boat wrt
earth
boat wrt
river
river wrt
earth
2 Cases
pointed perpendicular
to stream
travels perpendicular
to stream
Example 3.9
An airplane 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
Example 3.10
An airplane capable of moving 200 mph in still air. A
wind blows directly from the North at 50 mph.
If the airplane accounts for the wind (by pointing
the plane somewhat into the wind) and flies directly
east relative to the ground.
What is his resulting ground speed?
In what direction is the nose of the plane pointed?
193.6 mph
14.5 deg. north of east