1. +
Two-Dimensional Motion and
Vectors Chapter 3 pg. 81-105

2. +
What do you think?
 How are measurements such as mass and
volume different from measurements such as
velocity and acceleration?
 How can you add two velocities that are in
different directions?

3. +
Introduction to Vectors
 Scalar- a quantity that has magnitude but
no direction
 Examples: volume, mass, temperature,
speed
 Vector
- a quantity that has both
magnitude and direction
 Examples:acceleration, velocity,
displacement, force

4. +
Vector Properties
 Vectors are generally drawn as arrows.
 Length represents the magnitude
 Arrow shows the direction
 Resultant - the sum of two or more
vectors
 Make sure when adding vectors that
 Youuse the same unit
 Describing similar quantities

5. +
Finding the Resultant Graphically
 Method
 Draw each vector in the proper
direction.
 Establish a scale (i.e. 1 cm = 2 m)
and draw the vector the appropriate
length.
 Draw the resultant from the tip of the
first vector to the tail of the last
vector.
 Measure the resultant.
 The resultant for the addition of
a + b is shown to the left as c.

6. +
Vector Addition
 Vectorscan be moved parallel
to themselves without changing
the resultant.
 the red arrow represents the
resultant of the two vectors

7. +
Vector Addition
 Vectorscan be added
in any order.
 The resultant (d) is the
same in each case
 Subtraction is simply
the addition of the
opposite vector.

8. Sample Resultant Calculation
 A toycar moves with a
velocity of .80 m/s across a
moving walkway that
travels at 1.5 m/s. Find the
resultant speed of the car.

9. +
3.2 Vector Operations

10. +
What do you think?
 What is one disadvantage of adding vectors by the graphical
method?
 Is there an easier way to add vectors?

11. +
Vector Operations
 Use a traditional x-y coordinate system as shown below
on the right.
 The Pythagorean theorem and tangent function can be
used to add vectors.
 More accurate and less time-consuming than the
graphical method

12. + Pythagorean Theorem and Tangent Function

13. +
Pythagorean Theorem and Tangent
Function
 We can use the inverse of the tangent
function to find the angle.
 θ= tan-1 (opp/adj)
 Another way to look at our triangle
d
 d2 =Δx2 + Δy2 Δy
θ
Δx

14. +
Example
 An archaeologist climbs the great pyramid
in Giza. The pyramid height is 136 m and
width is 2.30 X 102m. What is the
magnitude and direction of displacement
of the archaeologist after she climbs from
the bottom to the top?

15. +
Example
 Given:
 Δy= 136m
 width is 2.30 X 102m for whole pyramid
 So, Δx = 115m
 Unknown:
d = ?? θ= ??

16. +
Example
 Calculate:  θ= tan-1 (opp/adj)
d2 =Δx2 + Δy2  θ= tan-1 (136/115)
 θ= 49.78°
d = √Δx2 + Δy2
d = √ (115)2 +(136)2
d = 178m

17. +
Example
 While following the directions on a
treasure map a pirate walks 45m north
then turns and walks 7.5m east. What
single straight line displacement could the
pirate have taken to reach the treasure?

18. +
Resolving Vectors Into Components

19. +
Resolving Vectors into Components
 Component: the horizontal x and vertical yparts that
add up to give the actual displacement
 Forthe vector shown at right, find the vector
components vx (velocity in the x direction) and vy
(velocity in the y direction). Assume that the angle is
35.0˚.
35°

20. +
Example
 Given: v= 95 km/h θ= 35.0°
 Unknown vx=??vy= ??
 Rearrange the equations
 sin θ= opp/ hyp
 opp=(sin θ) (hyp)
 cosθ= adj/ hyp
 adj= (cosθ)(hyp)

21. +
Example
vy=(sin θ)(v) vx= (cosθ)(v)
 vy= (sin35°)(95)  vx = (cos 35°)(95)
 vy= 54.49 km/h  vx = 77.82 km/h

22. +
Example
 Howfast must a truck travel to stay
beneath an airplane that is moving 105
km/h at an angle of 25° to the ground?

23. +
3.3 Projectile Motion

24. +
What do you think?
 Suppose two coins fall off of a table simultaneously. One
coin falls straight downward. The other coin slides off the
table horizontally and lands several meters from the base
of the table.
 Which coin will strike the floor first?
 Explain your reasoning.
 Would your answer change if the second coin was moving
so fast that it landed 50 m from the base of the table? Why
or why not?

25. +
Projectile Motion
 Projectiles: objects that are launched into the air
 tennis balls, arrows, baseballs, javelin
 Gravity affects the motion
 Projectile motion:
 The curved path that an object follows when
thrown, launched or otherwise projected near the
surface of the earth

26. +
Projectile Motion
 Pathis parabolic if air resistance is
ignored
 Path is shortened under the effects of air
resistance

27. Components of Projectile Motion
 As the runner launches
herself (vi), she is
moving in the x and y
directions.

28. +
Projectile Motion
 Projectile
motion is free fall with an initial
horizontal speed.
 Vertical
and horizontal motion are
independent of each other.
the acceleration is constant (-10 m/s2 )
 Vertically
 We use the 4 acceleration equations
 Horizontally the velocity is constant
 We use the constant velocity equations

29. +
Projectile Motion
 Components are used to solve for vertical
and horizontal quantities.
 Timeis the same for both vertical and
horizontal motion.
 Velocity at the peak is purely horizontal
(vy= 0).

30. +
Example
 The Royal Gorge Bridge in Colorado rises
321 m above the Arkansas river. Suppose
you kick a rock horizontally off the bridge
at 5 m/s. How long would it take to hit the
ground and what would it’s final velocity
be?

31. +
Example
 Given: d = 321m a = 10m/s2
vi= 5m/s t = ?? vf = ??
 REMEMBER we need to figure out :
 Up and down aka free fall (use our 4
acceleration equations)
 Horizontal (use our constant velocity
equation)

32. +
Classroom Practice Problem
(Horizontal Launch)
 People in movies often jump from buildings into
pools. If a person jumps horizontally by running
straight off a rooftop from a height of 30.0 m to a
pool that is 5.0 m from the building, with what
initial speed must the person jump?
 Answer: 2.0 m/s

33. +
Projectiles Launched at an Angle
 We will make a triangle and use our sin,
cos, tan equations to find our answers
 Vy = V sin θ
 Vx = V cosθ
 tan = θ(y/x)

34. +
Classroom Practice Problem
(Projectile Launched at an Angle)
 A golferpractices driving balls off a cliff and into
the water below. The edge of 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?
 Answer: 1.7 x 102m (170 m)