Physics 350

1
Physics 350

• Chapter 3
• Vectors and Two Dimensional Motion

2
Vectors

• Motion in 1D
• Negative/Positive for Direction
• Displacement
• Velocity
• Motion in 2D and 3D
• 2 or 3 displacements
• Too much work
• Easier way to describe these motions
• Vectors!
• Magnitude and direction
• Scalars magnitude only

3
Vectors and Scalars

1. my velocity (3 m/s)
2. my acceleration downhill (30 m/s2)
3. my destination (the lab - 100,000 m east)
4. my mass (150 kg)

4
Vectors

• A vector is composed of a magnitude and a
  direction
• Examples displacement, velocity, acceleration
• Magnitude of A is designated A or A
• Usually vectors include units (m, m/s, m/s2)
• A vector has no particular position
• (Note the position vector reflects displacement
  from the origin)

5
Comparing Vectors and Scalars

• A scalar is an ordinary number.
• A magnitude without a direction
• May have units (kg) or be just a number
• Usually indicated by a regular letter, no bold
  face and no arrow on top.
• Note the lack of specific designation of a
  scalar can lead to confusion

A
B
6
Vectors and their Properties

• Equality of Two Vectors
• Two vectors are equal if they have the same
  magnitude and the same direction
• Displacement can be the same for many paths
• Movement of vectors in a diagram
• Any vector can be moved parallel to itself
  without being affected

7
Vectors and their Properties (cont.)

• Negative Vectors
• Two vectors are negative if they have the same
  magnitude but are 180 apart (opposite directions
• A -B A (-A) 0
• Resultant Vector
• The resultant vector is the sum of a given set of
  vectors
• R C D

8
Vectors and their Properties (cont.)

• Adding Vectors
• Geometrically
• Scale drawings
• Triangle Method
• Algebraically
• More convenient
• Adding components of a vector

9
Vectors and their Properties (cont.)

• Adding Vectors Geometrically

• Draw the first vector with the appropriate length
  and in the direction specified
• Draw the second vector with the appropriate
  length and direction with its tail at the head of
  the first vector
• Construct the resultant (vector sum) by drawing a
  line from the tail of the first to the head of
  the second

10
Vectors and their Properties (cont.)

• When you have many vectors, just keep repeating
  the process until all are included
• The resultant is still drawn from the origin of
  the first vector to the end of the last vector

11
Vectors and their Properties (cont.)

• Vectors obey the Commutative Law of Addition
• The order in which the vectors are added doesnt
  affect the result
• A B B A

12
Vectors and their Properties (cont.)

• Vector Subtraction
• Same method as vector addition
• A B A (-B)

13
Vectors and their Properties (cont.)

• Scalar multiplication
• Scalar x Vector
• Change magnitude or direction of vector
• Applies to division also

A
3A
-1A
(1/3)A
14
Components of a Vector

• Vectors can be split into components
• x direction component
• y direction component
• z component too! But we wont need it this
  semester!
• A vector is completely described by its
  components
• Choose coordinates
• Rectangular

15
Components of a Vector (cont.)

• Components of a vector are the projection along
  the axis
• Ax A cos ?
• Ay A sin ?
• Then, Ax Ay A
• Looks like Trig
• Because it is!
• cos ? Ax / A

16
Components of a Vector (cont.)

• Hypotenuse is A (the vector)
• Magnitude is defined by Pythagorean theorem
• v(Ax2 Ay2) A
• Direction is angle
• tan ? Ay / Ax
• Equations valid if ? is respect to x-axis
• Use components instead of vectors!

17
Components of a Vector (cont.)

• Adding Vectors Algebraically
• Draw the vectors
• Find the x and y components of all the vectors
• Add all the x components
• Add all the y components
• If R A B,
• Then Rx Ax Bx
• and Ry Ay By
• Use Pythagorean Theorem to find magnitude and
  tangent relation for angle

18
Components of a Vector (cont.)

• Example
• A golfer takes two putts to get his ball into
  the hole once he is on the green. The first ball
  displaces the ball 6.00 m east, the second 5.40 m
  south. What displacement would have been needed
  to get the ball into the hole on the first putt?

19
Components of a Vector (cont.)

• Example
• A hiker begins a trip by first walking 25.0km
  southeast from her base camp. She then walks
  40.0km, 60.0 north of east where she finds the
  forest rangers tower.
• Find the components of A
• Find the components of B
• Find the components of the resultant vector R A
  B
• Find the magnitude and direction of R

20
Displacement, Velocity, and Acceleration in Two
Dimensions

• So, why all this time to study vectors?
• We can more describe 2D motion (3D too) more
  generally
• We can apply to it to a variety of physical
  problems
• Force
• Work
• Electric Field
• Displacement, Velocity, and Acceleration

21
Motion in Two Dimensions

• Two dimensional motion under constant
  acceleration is known as projectile motion.
  Projectile path is known as trajectory.
• Trajectory can fully be described by equations
• QM cannot be fully described
• Motion in x direction and y direction are
  independent of each other

22
Motion in 2D (cont.)

• Assumptions
• Ignore air resistance
• Ignore rotation of the earth (relative)
• Short range, so that g is constant
• Object in 2D motion will follow a parabolic path
• Ball throw

23
Projectile Motion

• Important points of projectile motion
• It can be decomposed as the sum of horizontal and
  vertical motions.
• The horizontal and the vertical components are
  totally independent of each other
• The gravitational acceleration is perpendicular
  to the ground so it affects only the
  perpendicular component of the motion.
• The horizontal component of the motion has zero
  acceleration, because the gravitational
  acceleration has no horizontal component and we
  neglect the air drag.

24
Motion in 2D

• Falling balls
• Both balls hit the ground at the same time
• Initial horizontal motion of yellow ball does not
  affect its vertical motion

25
Motion in 2D

• Projectile motion can be decomposed as the sum of
  vertical and horizontal components

26
Motion in 2D (cont.)

• Projectile Motion
• x direction uniform motion
• ax 0
• y direction constant acceleration
• ay -g
• Initial velocity can be broken into components
• v0x v0cos?0
• v0y v0sin ?0

27
Motion in 2D (cont.)

• Projectile Motion
• Complimentary initial angles results in the same
  range
• Maximum range?
• 45º

28
Motion in 2D (cont.)

• x direction motion
• ax 0
• v0x v0cos?0 vx constant
• Since velocity is constant, a 0
• So the only useful equation is, ? x vxot
• ?x v0cos?0 t
• From our four equations of motion, this one is
  only operative equation with uniform velocity (no
  acceleration)

29
Motion in 2D (cont.)

• y direction motion
• v0y v0sin ?0
• free fall problem
• a -g
• take the positive direction as upward
• uniformly accelerated motion, so all the motion
  equations from Chapter 2 hold
• v v0sin ?0 - gt
• ?y v0sin ?0 t - ½ gt2
• v2 (v0sin ?0)2 - 2g ?y
• Not so concerned about the average velocity in
  projectile problems

30
Concept Test

• You drop a package from a plane flying at
  constant speed in a straight line. Without air
  resistance, the package will
• 1) quickly lag behind the plane while falling
• 2) remain vertically under the plane while
  falling
• 3) move ahead of the plane while falling
• 4) not fall at all

31
Concept Test

• 2) remain vertically under the plane while falling

32
ConcepTest Firing Balls I

• A small cart is rolling at constant velocity on
  a flat track. It fires a ball straight up into
  the air as it moves. After it is fired, what
  happens to the ball?

1) it depends on how fast the cart is moving 2)
it falls behind the cart 3) it falls in front of
the cart 4) it falls right back into the cart 5)
it remains at rest
33
ConcepTest Firing Balls I

• A small cart is rolling at constant velocity on
  a flat track. It fires a ball straight up into
  the air as it moves. After it is fired, what
  happens to the ball?

1) it depends on how fast the cart is moving 2)
it falls behind the cart 3) it falls in front of
the cart 4) it falls right back into the cart 5)
it remains at rest
In the frame of reference of the cart, the ball
only has a vertical component of velocity. So it
goes up and comes back down. To a ground
observer, both the cart and the ball have the
same horizontal velocity, so the ball still
returns into the cart.
http//www.youtube.com/watch?vFLUOgO2-0lA
34
Movie

• Monkey and the Hunter
• Will the ball go over or under the monkey?
• http//www.youtube.com/watch?vcxvsHNRXLjw
• What if I adjust the speed?
• http//www.youtube.com/watch?vnwQwk15TAh4

35
Motion in 2D (cont.)

• Example
• A stone is thrown upward from the top of a
  building at an angle of 30.0 to the horizontal
  and with an initial speed of 20.0 m/s. The point
  of release is 45.0m above the ground.
• Find the time of flight
• Find the speed at impact
• Find the horizontal range of the stone

36
Motion in 2D (cont.)

• Example
• An artillery shell is fired with an initial
  velocity of 300 m/s at 55.0 above the
  horizontal. To clear the avalanche, it explodes
  on the mountainside 42.0 s after firing. What
  are the x- and y-coordinates of the shell where
  it explodes, relative to its firing point?

37
Motion in 2D (cont.)

• Summary of Projectile Motion
• Assuming no air resistance, x-direction velocity
  is constant
• y-direction is similar to free fall problem
• velocity, displacement
• x-direction and y-direction are independent of
  each other