Two-Dimensional Motion
and Vectors
Chapter 3
Introduction to Vectors
• A scalar is a quantity that can be completely
specified by its magnitude
• A vector is a physical quantity that has both
magnitude and direction.
• Some examples of vector quantities are
velocity, acceleration, displacement and
forces.
• Vectors can be added graphically, but when
adding vectors you must be sure that they
have the same units and that the vectors
describe similar quantities.
• The answer found by adding vectors is
Vector operations
The magnitude and direction of the resultant of
two perpendicular vectors can be calculated
using the Pythagorean Theorem and the
tangent function.
Magnitude of vector → Pythagorean Theorem
Direction of vector → Tangent function
Basic Vector Operations
Both a magnitude and a direction must be specified for a vector quantity, in
contrast to a scalar quantity which can be quantified with just a number.
Any number of vector quantities of the same type (i.e., same units) can be
combined by basic vector operations.
Addition of 2 Vectors
Addition of 3 Vectors
Set # 9
P/3
A football player runs 15 m down the field, and then turns to the left at an
angle of 15º from his original direction and running an additional 10 m
before getting tackled
Find the magnitude and direction
By
15º
Bx
15 m
P/5 Set # 9
A person walk’s the path shown below, what’s is the person
resultant displacement measured from the starting person
100 m
Resultant Vector
?
300 m
30°
60°
Ax = ______
Ay = ______
Bx = ______
By = ______
Cx = ______
Cy = ______
Dx = ______
Dy = ______
∑x = ______
∑y = ______
Example 4:
A plane flies from city A to city B. City B is 1,540 km west
and 1,160 km south of city A.
What is the total displacement and direction of the plane?
a)
b)
c)
d)
1,930 km, 43° south of west
1,850 km, 43° south of west
1,850 km, 37° south of west
1,930 km, 37° south of west
A plane flies from city A to city B. City B is 1,540 km west
and 1,160 km south of city A.
What is the total displacement and direction of the plane?
A
1,160 km
B
Example # 5
During a rodeo, a clown runs 8 m north, turns 35º east of north and
runs 3.5 m. Then after awaiting for the bull to come near, the clown
turns due east and runs 5m to exit the arena.
Find the clown’s displacement
5m
35º
8m
Resultant Vector
Ax = _____
Ay = _____
Bx = _____
By = _____
Cx = _____
Cy = _____
∑x = _____
∑y = _____
Example 6:
A duck waddles 2.5 m east and 6 m north.
What is the magnitude and direction of the duck’s
displacement with respect to its original position?
a)
b)
c)
d)
3.5 m at 19° north of east
6.3 m at 67° north of east
6.5 m at 67° north of east
6.5 m at 72° north of east
Physics
Set # 10
Student Name: ____________________________________
P/1
Oct/2/2007
Class Period: _______
A person walks the path shown below:
a)
Find the person’s resultant displacement measured from the starting point
b)
Determine the direction
20º
60º
Answers: a)__________
30º
150 m
P/2
500 m
b) ________
A person walks the path shown below:
a)
Find the person’s resultant displacement measured from the starting point
b)
Determine the direction
250 m
Answers: a)_________
25º
b) _________
15º
Physics
Set # 11
Student Name: ____________________________________
P/1
Oct/3/2007
Class Period: _______
Mr. C designed play 2244 in football that will send a receiver down the field as explained in the following
diagram.
a)
Find the displacement of this receiver
b)
Find the direction
30º
6 yds
50º
14 yds
P/2
Answers: a) _________
b) _________
Mr. C designed play 4877 in football that will send a receiver down the field as explained in the following
diagram.
a)
Find the displacement of this receiver
12 yds
b)
Find the direction
4 yds
20º
10 yds
Answers: a) _________
b) ________
Physics
Set # 12
Student Name: ____________________________________
P/1
Oct/4/2007
Class Period: _______
Mr. C designed play 485 in football that will send a receiver down the field as explained in the following
diagram.
a)
Find the displacement of this receiver
b)
Find the direction
20º
40º
4 yds
60º
Answers: a) _________
b) _________
16 yds
P/2
Mr. C designed play 4877 in football that will send a receiver down the field as explained in the following
diagram.
11 yds
a)
Find the displacement of this receiver
b)
Find the direction
75º
40º
15 yds
Answers: a) _________
b) _________
Physics
Set # 13
Student Name: ____________________________________
P/1
Oct/4/2007
Class Period: _______
Mr. C designed play 775 in football that will send a receiver down the field as explained in the
following diagram.
a)
Find the displacement of this receiver
b)
Find the direction
40º
4 yds
60º
12 yds
P/2
Answers: a) _________
b) _________
Mr. C designed play 4877 in football that will send a receiver down the field as explained in the
following diagram.
8 yds
a)
Find the displacement of this receiver
b)
Find the direction
65º
40º
12 yds
Answers: a) _________
b) _________
P/3
A person walks the path shown below:
a)
Find the person’s resultant displacement measured from the starting point
b)
Determine the direction
20º
Answers: a)__________
60º
b) ________
30º
180 m
P/4
A person walks the path shown below:
a)
Find the person’s resultant displacement measured from the starting point
b)
Determine the direction
Answers: a)_________
250 m
b) _________
25º
15º
Vectors
• The study of vectors allows for breaking a single vector into components:

the vertical component

the horizontal component.
•
By breaking a single vector into two components, or resolving it into its
components, an object’s motion can sometimes be described more
conveniently in terms of directions, such as north to south, or up and down.
vi
vy
vx
Essay: Wernher von Braun
2 pages/ 12 font
Chunk Paragraph:
topic sentence + 2 concrete details +4 commentaries
+ concluding sentence
•
•
•
•
•
•
The early childhood
Family
His achievements in Germany
His involvement in the Nazi Party
How did he ended up in U.S.A.?
His achievements in U.S.A.
Projectile Motion
Projectile Motion
• A projectile is any object upon which the only force acting is
gravity.
• Projectiles travel with a parabolic trajectory due to the
influence of gravity,
• There are no horizontal forces acting upon projectiles and thus
no horizontal acceleration,
• The horizontal velocity of a projectile is constant (never
changing in value),
• There is a vertical acceleration caused by gravity; its value is
-9.81 m/s², down.
• The vertical velocity of a projectile changes by – 9.81 m/s
each second.
• The horizontal motion of a projectile is independent of its
vertical motion.
Projectile Motion
A projectile is an object upon which the only force acting is gravity.
There are a variety of examples of projectiles
1. An object dropped from rest is a projectile (provided that the
influence of air resistance is negligible).
2. An object which is thrown vertically upwards is also a projectile
(provided that the influence of air resistance is negligible).
3. An object which is thrown upwards at an angle is also a projectile
(provided that the influence of air resistance is negligible).
•
A projectile is any object which once projected continues in motion
by its own inertia and is influenced only by the downward force of
gravity.
Projectile Motion
• By definition, a projectile has only one force acting upon
- the force of gravity. If there were any other force acting
upon an object, then that object would not be a
projectile. Thus, the free-body diagram of a projectile
would show a single force acting downwards and labeled
"force of gravity" (or simply Fg).
• This is to say that regardless of whether a projectile is
moving downwards, upwards, upwards and rightwards,
or downwards and leftwards, the free-body diagram of
the projectile is still as depicted in the following diagram
Projectile Motion
Projectile Motion
Projectile Motion
Projectile Motion
Many projectiles not only undergo a vertical motion, but
also undergo a horizontal motion.
That is, as they move upward and/or downward they are
also moving horizontally.
There are the two components of the projectile's motion horizontal and vertical motion. And since "perpendicular
components of motion are independent of each other,"
these two components of motion can (and must) be
discussed separately.
• The goal of this part of the lesson is to discuss the
horizontal and vertical components of a projectile's
motion; specific attention will be given to the
presence/absence of forces, accelerations, and velocity.
Projectile Motion
Projectile Motion
• In this course, we are going to study 2
different situations about projectiles:
• 1. Projectiles launched horizontally
• 2. Projectiles launched at an angle
θ
Projectiles launched horizontally
Horizontal motion of a projectile
∆x = vx ∙∆t
vx = vi,x = constant
Vertical Motion of a projectile
∆y = ½ a(∆t)²
vy,f = a ∆t
vy,f ² = 2 a∆y
Projectiles launched at an angle
v
vx
Ө
vy
vx,i = vi cos Ө
vy,i = vi sin Ө
Horizontal motion of a projectile
∆x = vi ∙cos θ∙∆t
vx =vi ∙cosθ = constant
Vertical Motion of a projectile
∆y = vi∙sinθ ∙∆t +½ a(∆t)²
vy,f = vi∙sin θ + a∙∆t
vy,f² = vi² (sin θ)² +2 a∙∆y
Example #1
The New York Yankees, in their quest for an unprecedented 27th.
World Series Championship, have an extraordinary pitching rotation.
One of our best pitchers is Mike Mussina,aka Moose, who is capable
of throwing his fastball at 92 mph.
If he pitched to the home-plate (horizontally) (60 ft away), how far will
the ball fall vertically by the time it reaches home plate?.
HORIZONTAL MOTION OF A
PROJECTILE
VERTICAL MOTION OF A
PROJECTILE
∆x = vix∙∆t
vix = constant
vix = vx
∆y = a (∆t)²
2
Example # 1
Moose, who is capable of throwing his fastball at 92 mph.
If he pitched to the home-plate (horizontally) (60 ft away), how far
will the ball fall vertically by the time it reaches home plate?.
Given: vix = 92 mi/h → vix = 41 m/s
∆x = 60 ft → ∆x = 18.29 m
Unknowns: ∆t =?, ∆y=?
Solution:
• First find the time that it takes for the baseball to reach
home plate:
∆x = vix∙∆t
∆t = ∆x/vix
∆t = 18.29 m / 41 m/s = 0.44 s
• Now find the vertical fall:
∆y = ½ a∙(∆t)² = ½ (-9.81 m/s²) (0.44 s)² = - 0.95 m
Example # 2
A plane wants to drop a cargo package into a drop area.
The plane is 120 m above the ground, and traveling at a velocity of
40 m/s in the positive x direction.
Find how long will it take the cargo load to reach the ground, and how
much horizontal distance will it take to drop it in the landing area.
• Given: ∆y = -120 m
• Unknowns: ∆t, ∆x
a= -9.81 m/s2
vx = 40 m/s
120 m
∆x
Drop
Zone
Example 2:
A plane wants to drop a cargo package into a drop area.
The plane is 120 m above the ground, and traveling at a velocity of
40 m/s in the positive x direction.
Find how long will it take the cargo load to reach the ground, and how
much horizontal distance will it take to drop it in the landing area.
• Given: ∆y = -120 m
a= -9.81 m/s2
vi,x = 40 m/s
•
vi,y = 0 m/s
• Unknowns: ∆t, ∆x
• Solution:
∆y = ½ a ∆t² → ∆t² = 2∆y/a
∆t² = (2) (-120 m)/ (- 9.81 m/s²)
∆t² = 24.464 → ∆t = √ 24.64
∆t = 4.96 s
∆x = vi,x ∙∆t
∆x = (40 m/s) (4.96 s)
∆x = 198.40 m
Example # 3
People in movies often jump from buildings into pools. If a person jumps from
a 10th. floor(30 m) to a pool that is 5 m away from the building, with what
initial horizontal velocity must the person jump?
30 m
5m
• Given:
• Δy = - 30 m
Δx = 5 m
a = - 9.81 m/s²
Example 4:
A zookeeper finds an escaped monkey hanging from a light pole.
Aiming his tranquilizer gun at the monkey, the zookeeper kneels 12 m from
the light pole which is 6 m high.
The tip of his gun is 1 m above the ground.
The monkey tries to trick the zookeeper by dropping a banana, then
continues to hold onto the light pole.
At the moment the monkey releases the banana, the zookeeper shoots.
If the tranquilizer dart travels at 50 m/s, will the dart hit the monkey or the
banana?
6m
1m
12 m
Steps
• 1. Find the angle :
Θ = tan‾¹ (5/12) = 23º
• 2. Find the time it takes to reach the target:
∆x = vi ∙cos θ∙∆t
∆t = ∆x / vi ∙cos θ
∆t =
Example 3:
A ball is fired from the ground with an initial speed of 1,700 m/s at an initial
angle of 30° to the horizontal.
Neglecting air resistance, find:
a) The ball’s horizontal range
b) The amount of time the ball was in motion
•
•
Given: vi = 1,700 m/s
Unknowns: ∆t, ∆x, ∆y
•
Steps:
Θ = 30°
a = -9.81 m/s²
∆x = vi cos Θ ∆t → ∆x = (1,700 m/s) cos 30°∙ ∆t
∆x = 1,472.24 m/s ∆t
∆y = vi sin Θ ∆t + ½ a (∆t)² →
∆y = (1,700 m/s) sin 30° ∆t + ½ (- 9.81 m/s²) ∆t²
∆y = 850 m/s ∆t – 4.91 ∆t²
∆y = 0 (because it ends at the same vertical position)
4.91 ∆t²- 850 ∆t = 0
4.91 ∆t- 850 = 0
∆t = 173 s
∆x = (1,472.24 m/s) (173 s)
∆x = 254,697.52 m
Golf and Physics
When you play golf, you have a variety of golf clubs to hit for different
distances. Most people use the driver to tee off on the par 4’s, and then a 7
iron from the middle of the fairway. The driver has typically an angle on
10˚, and the 7 iron an angle of 35˚,
Calculate the horizontal and vertical displacement of a golfer that hits golf
balls at an average initial velocity of 120 mi/h for the initial 10 s after
impact. www,explore science.com
Y axis
Height
X axis
(0,0)
Time
(t,0)
Driver 10˚
vi = 120 mi/h (53.63 m/s)
∆t
∆x = vi cosθ ∆t
∆y = vi sin θ ∆t + ½ a ∆t ²
1s
52.82 m
4.41 m
2s
105.63 m
- 1.00 m
3s
158.45 m
4s
211.26 m
5s
264.08 m
6s
316.89 m
7s
369.71 m
8s
417.22 m
9s
469.37 m
10s
521.52 m
7 iron 35˚
vi = 120 mi/h (53.63 m/s)
∆t
∆x = vi cosθ ∆t
∆y = vi sin θ ∆t + ½ a ∆t ²
1s
43.93 m
25.86 m
2s
87.86 m
41.90 m
3s
131.79 m
48.14 m
4s
175.72 m
44.56 m
5s
219.66 m
31.18 m
6s
263.59 m (241 yd)
7.99 m
7s
307.52 m
- 25.02 m
8s
351.45 m
9s
395.38 m
10s
439.31 m
Something’s wrong???
• The average displacement for a 7 iron is
150 yds!!
• Calculate the angle for:
• Vi = 120 mi/h → 53.63 m/s
• ∆x = 150 yds → 137.16 m
• ∆t = 6.3s ?
• θ = ?????
BONUS POINTS
The great pyramid of Giza is approximately 150 m high
and has a square base approximately 230 m on a side.
What is the approximate area of a horizontal cross
section of the pyramid taken 50 m above its base?
(Determine this by using 3 different methods!!)
a) 5,880 m²
b) 11,760 m²
c) 23,510 m²
d) 35,270 m²
A
B
C
F
G
E
D
CD = 230 m
AG = 150 m
GF = 50 m
A
B
C
F
G
E
D
CD = 230 m
AG = 150 m
GF = 50 m
AF = 100 m
A
AG = AF
CD BE
100 m
BE = 153.33 m
BE² = 23,511 m²
B
F
E
50 m
C
G
230 m
D
BONUS POINTS
Using a protractor, a student measures the sum of the interior angles
in a triangle and obtains 176°.
What is the percent error of this measurement?
a)
b)
c)
d)
0.04%
2.22%
2.27%
4.00%
BONUS POINTS
For what value of c will the function below have exactly one vertical
asymptote?
y = ____4______
x² – cx + 9
a)
b)
c)
d)
0
4
6
9
Comparing a Driver and a 7 iron
150 m
∆y (m)
100 m
75 m
100
50
25
∆x (m)
Example # 3
From the top of a tall building, a gun is fired.
The bullet leaves the gun at a speed of 340 m/s, parallel to the ground.
The bullet puts a hole in a window of another building and hits the wall
that faces the window. (y = 0.60 m, and x = 6.8 m.)
Using the data in the drawing, determine the distances D and H, which
locate the point where the gun was fired.
Assume that the bullet does not slow down as it passes through the
window
Example 4:
A golfer practices driving balls of a cliff and into water below.
The cliff is 15 m above the water.
If the golf ball is launched at 51 m/s at an angle of 15°.
How far does the ball travel horizontally before hitting the water?
15 m
• For
ax² + bx + c = 0
the value of x is given by:
Summer School Physics
Set # 9
Student Name: _____________________________________
P/1
June/16/2008
Class Period: ______
A golfer practices driving balls of a cliff and into water below.
The cliff is 12 m above the water.
If the golf ball is launched at 61 m/s at an angle of 25°.
How far does the ball travel horizontally before hitting the water?
Answer: ___________
P/2
A place kicker must kick a football from a point 36 m (about 40 yd) from the goal, and the ball
must clear the crossbar which is 3.05 m high.
When kicked, the ball leaves the ground with a speed of 20 m/s at an angle of 53° to the
horizontal.
A. By how much does the ball clear of fall short of clearing the crossbar?
B. Does the ball approach the crossbar while still rising or while falling
Answers: A = ___________
B = ___________
P/3
A daredevil jumps a canon 12 m wide. To do so, he drives a car up a 15° incline.
A) What minimum speed must he achieve to clear the canyon?
B) If the daredevil jumps at this minimum speed, what will be his speed when he
reaches the other side?
Answers: A = ___________
B = ___________
Example 4:
A golfer practices driving balls of a cliff and into water below.
The cliff is 15 m from the water.
If the golf ball is launched at 51 m/s at an angle of 15°.
How far does the ball travel horizontally before hitting the water?
• Given: ∆y = - 15 m, v = 51 m/s, Ө = 15°
• Solution:
• 1. Find the time that it takes the golf ball to hit the water
• 2. Find the horizontal displacement
• To find the time:
∆y = vi sinӨ ∆t + ½ a ∆t²
- 15 =(51 m/s) sin15°∆t - ½ (9.81) ∆t²
•
- 15 = 13.2 ∆t – 4.91 ∆t²
4.91 ∆t² 13.2 ∆t – 15 = 0
∆t = - b ±√b² 4 ac
2a
∆t = 13.20 ± 21.65
∆t = 3.55 s
∆x = vi cosӨ∙∆t
∆x = 174.87 m
Example # 5
A place kicker must kick a football from a point 36 m (about 40 yd) from
the goal, and the ball must clear the crossbar which is 3.05 m high.
When kicked, the ball leaves the ground with a speed of 20 m/s at an
angle of 53° to the horizontal.
A. By how much does the ball clear of fall short of clearing the
crossbar?
B. Does the ball approach the crossbar while still rising or while falling?
∆y = 3.05 m
Ө = 53°
∆x = 36 m
A place kicker must kick a football from a point 36 m (about 40 yd)
from the goal, and the ball must clear the crossbar which is 3.05 m
high.
When kicked, the ball leaves the ground with a speed of 20 m/s at an
angle of 53° to the horizontal.
•
•
•
•
•
∆x = 36 m,, Ө = 53°, a = - 9.81 m/s², vi = 20 m/s
∆y = 3.05 m ?
Unknowns: ∆t
Solution:
First find the time that it takes for the ball to reach the goal post, and
then determine how high will the ball be at that time to see if it clears
the goal post.
• ∆x = vi cosӨ ∆t →
• 36 m = (20 m/s) (cos 53°) ∆t
• 36 m = (12.04 m/s) ∙∆t
• ∆t = 36 m/ 12.04 m/s
• 2.99 s
Given:
Example # 6:
A daredevil jumps a canon 12 m wide. To do so, he drives a car up a 15°
incline.
A) What minimum speed must he achieve to clear the canyon?
B) If the daredevil jumps at this minimum speed, what will be his speed
when he reaches the other side?
Physics
Set # 14
Student Name: ____________________________________
P/1
October 30, 2007
Class Period: _______
If you throw a baseball horizontally at 85 mph to home plate (60 ft away), how much will
the baseball drop, once it reaches home plate (neglecting air resistance)?
Answer: ____________
P/2
If you throw a baseball horizontally at 75 mph to home plate (60 ft away), how much will
the baseball drop, once it reaches home plate (neglecting air resistance)?
Answer: ____________
P/3
A plane wants to drop a cargo package into a drop area. The plane is 150 m above the
ground, and traveling at a velocity of 50 m/s in the positive x direction.
Find how long will it take the cargo load to reach the ground, and how much horizontal
distance will it take to drop it in the landing area.
Answer: ___________
P/4
A plane wants to drop a cargo package into a drop area. The plane is 250 m above the
ground, and traveling at a velocity of 80 m/s in the positive x direction.
Find how long will it take the cargo load to reach the ground, and how much horizontal
distance will it take to drop it in the landing area.
Answer: ___________
Physics
Set # 15
Student Name: ____________________________________
P/1
A ball is fired from the ground with an initial speed of 500 m/s at an initial angle of 40° to
the horizontal. Neglecting air resistance, find:
a) The ball’s horizontal range
b) The amount of time the ball was in motion
c) The maximum height that the ball reaches
P/2
Answer: ___________
Answer: ___________
Answer: ___________
A ball is fired from the ground with an initial speed of 400 m/s at an initial angle of 60° to
the horizontal. Neglecting air resistance, find:
a) The ball’s horizontal range
b) The amount of time the ball was in motion
c) The maximum height that the ball reaches
P/3
November 2, 2007
Class Period: _______
Answer: ___________
Answer: ___________
Answer: ___________
A golfer practices driving balls of a cliff and into water below. The cliff is 15 m from the
water. If the golf ball is launched at 60 m/s at an angle of 10°.
a) How far does the ball travel horizontally before hitting the water?
Answer: __________
b) What was the maximum height of the ball while in the air?
Answer: __________
∆y = ½ a (∆t)²
________
∆t = √ 2 ∆y
a
∆t = 4.95 s
∆x = Vx∆t = (40 m/s) (4,95 s) = 198 m
Bonus Points
• A fifth-grade class is using pattern blocks in the shape of congruent
equilateral triangles to devise and solve problems involving fractions. One
group devises the problem illustrated below.
• Given that the sum of Shapes A and B represents 5/8, which of the
following represent 1 ¼?
• A)
Projectile Motion
Physics
Driver 10˚
vi = 120 mi/h
∆t
1s
2s
3s
4s
5s
6s
7s
8s
9s
10s
∆x = vi cosθ ∆t
∆y = vi sin θ ∆t + ½ a ∆t ²
7 iron 35˚
vi = 120 mi/h
∆t
1s
2s
3s
4s
5s
6s
7s
8s
9s
10s
∆x = vi cosθ ∆t
∆y = vi sin θ ∆t + ½ a ∆t ²