Projectile (2D) Motion
AP Physics B
September 21, 2010
Trigonometry Review
• Application of Trigonometry to Vectors
Trigonometry
R
y
q
x
y
sin q 
R
x
cos q 
R
y
tan q 
x
y = R sin q
x = R cos q
R2 = x2 + y2
Example 1: Find the height of a building
if it casts a shadow 90 m long and the
indicated angle is 30o.
The height h is opposite 300 and
the known adjacent side is 90 m.
opp
h
tan 30 

adj 90 m
0
h
300
h = (90 m) tan 30o
90 m
h = 51.96 m
Finding Components of Vectors
A component is the effect of a vector along
other directions. The x and y components of
the vector (R,q) are illustrated below.
x = R cos q
R
q
x
y
y = R sin q
Finding components:
Polar to Rectangular Conversions
Example 2: A person walks 400 m in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
N
R
q
x
400 m
y
30o
E
y=?
x=?
The x-component (E) is ADJ:
x = R cos q
The y-component (N) is OPP:
y = R sin q
E
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
Note: x is the side
400 m
30o
y=?
x=?
E
x = (400 m) cos 30o
= +346 m, E
adjacent to angle 300
ADJ = HYP x Cos 300
x = R cos q
The x-component is:
Rx = +346 m
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
Note: y is the side
400 m
30o
y=?
x=?
E
opposite to angle 300
OPP = HYP x Sin 300
y = R sin q
y = (400 m) sin 30o
The y-component is:
= + 200 m, N
Ry = +200 m
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
400 m
30o
Rx =
Ry =
+200 m
E
The x- and ycomponents are
each + in the
first quadrant
+346 m
Solution: The person is displaced 346 m east
and 200 m north of the original position.
Components of Motion
• You can use the same approach to describe motion—
and the motion doesn’t have to be in straight lines.
• Using vectors will allow you to analyze the behavior of
batted balls, planets circling the Sun, and even
electrons in atoms.
• Think of ball moving in x- and y-directions
simultaneously.
• That is, it has a velocity in the x-direction (vx) and a
velocity in the y-direction (vy) at the same time.
• The combined velocity components describe the
actual motion of the ball.
Components of Motion
• If a constant velocity (v) in a direction at an
angle (Θ) relative to the x-axis is given,
then the velocities in the x- and y- directions
are obtained by resolving, or breaking
down, the velocity vector into components
of motion:
vx = v cos Θ
vy = v sin Θ
Ex. Shooting Pool
Shooting a game of pool, you hit a ball that causes
it to move diagonally with a constant velocity of
0.50 m/s at an angle of 37° relative to the x-axis.
Find how far it travels in 3.0 s by using x- and ycomponents of motion.
Projectile Motion
• A familiar example of two-dimensional, curvilinear
motion is the motion of objects thrown or projected
by some means (cannon, baseball bat, etc.)
• Projectile Motion is considered to be in free fall, so
the only acceleration of a projectile is due to gravity.
• We can use vector components to analyze projectile
motion. We simply break up the motion into its xand y-components and treat them separately.
Horizontal Projection
A ball is projected from a height of 25.0 m
above the ground and is thrown with an initial
horizontal velocity of 8.25 m/s (fig 3.16)
(a) How long is the ball in flight before striking
the ground?
(b) How far from the building does the ball strike
the ground?
Teeing Off
A golf ball is hit off the tee with an initial
velocity of 30.0 m/s at an angle of 35° to
the horizontal (fig 3.17)
(a) What is the maximum height reached by
the ball?
(b) What is its range?
Hit or Miss?
A young girl standing on a bridge throws a stone
with an initial velocity of 12 m/s at a downward
angle of 45° to the horizontal, in an attempt to
hit a block of wood floating in the river below.
If the stone is thrown from a height of 20 m and
it reaches the river when the block is 13 m from
the bridge, does the stone hit the block?
Homework:
• Read Conceptual Physics (Hewitt)
Chapters 2 & 3
• Do Problems 19—28
• From College Physics (Wilson) Do
Problems…9, 15, 73, 74, 79, 83, 84, 85,
89, 91
Projectile Practice
1. A ball is thrown straight up with
a speed of 12.5 m/s.
(a) How high does it go and
(b) how much time does it take to
get there?
2. A volkswagon runs straight off a
cliff traveling at a speed of 34.5 m/s.
If the cliff is 12.5 m high, how far
horizontally does the car travel
before it smashes into the ground?
3. A stealth bomber on a training
mission drops one of its bombs from
a height of 3,500m. The bomb
travels a horizontal distance of 1.25
km. What was the plane’s horizontal
speed?
4. An arrow is launched with a
velocity of 88.7 m/s at an angle
of 33.0° to the horizontal. How
far does the arrow travel?
5. A brick is thrown upward from the
top of a building at an angle of 25°
with an initial speed of 15 m/s. It
strikes the ground below. If the brick
is in flight for 3.0s, how tall is the
building?
6. A ball is thrown at an angle of 43°
to the horizontal. It travels a distance
of 75m in 2.3s.
(a) What was its original velocity?
(b) How high did it go?
Homework:
• Read Conceptual Physics (Hewitt)
Chapters 2 & 3
• Do Problems 19—28
• From College Physics (Wilson) Do
Problems…9, 15, 73, 74, 79, 83, 84, 85,
89, 91
Monkey and the Hunter
A zookeeper finds an escaped monkey
hanging from a 4-meter high tree eating a
banana. Aiming her tranquilizer gun at the
monkey, the zookeeper kneels and aims
21.8° from the ground, 10 meters from the
tree. The monkey thinks he is smart; as
soon as the zookeeper fires her gun, the
monkey releases from the tree. If the
tranquilizer dart travels at 50.0m/s, will the
zookeeper hit the monkey?