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Projectile (2D) MotionAP Physics B
September 21, 2010
AP Physics B
September 21, 2010
Trigonometry Review• Application of Trigonometry to
Vectors
y
x
R
q
y = R sin q y = R sin q
x = R cos qx = R cos q
siny
R
cosx
R
tany
x R2 = x2 +
y2
R2 = x2 + y2
Trigonometry
Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o.
90 m
300
The height h is opposite 300
and the known adjacent side is 90 m.
h
h = (90 m) tan 30o
h = 51.96 mh = 51.96 m
0tan 3090 m
opp h
adj
Finding Components of VectorsA component is the effect of a vector along other directions. The x and y components of the vector (R, )q are illustrated below.
x
yR
q
x = R cos q
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?
x
yR
q
x = ?
y = ?400 m
30o
E
N
The y-component (N) is OPP:
The x-component (E) is ADJ:
x = R cos qy = R sin q
E
N
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?
x = R cos q
x = (400 m) cos 30o
= +346 m, E
x = ?
y = ?400 m
30o
E
N Note: x is the side adjacent to angle
300
ADJ = HYP x Cos 300
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?
y = R sin q
y = (400 m) sin 30o
= + 200 m, N
x = ?
y = ?400 m
30o
E
N
OPP = HYP x Sin 300
The y-component is:Ry = +200 m
Note: y is the side opposite to angle
300
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?
Rx = +346 m
Ry = +200 m
400 m
30o
E
NThe x- and y- components are each + in
the first quadrant
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 y- components 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 x- and 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?
