Chapter 3 Motion in two or three dimensions
Transcript of Chapter 3 Motion in two or three dimensions
Chapter 3
Motion in two or three dimensions
Lecture by Dr. Hebin Li
PHY 2048, Dr. Hebin Li
Announcements
As requested by the Disability Resource Center:
In this class there is a student who is a client of Disability Resource Center. Would
someone please volunteer to share his or her notes with that student for the semester?
If willing, please meet with me after this class. Then, visit Disability Resource
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retrieve. Or, Disability Resource Center can provide special carbonless (NCR) paper
that provides immediate copies of your notes. Thank you.
Lecture slides (for Monday class) and recitation outline (for Wednesday class) can
be downloaded at http://faculty.fiu.edu/~hebli/teaching/phy-2048-schedule/
Pop quizzes can be in either lectures (Monday) or recitations (Wednesday). Since
they are random, I cannot tell you the exact dates.
PHY 2048, Dr. Hebin Li
Due at 11:59pm on Sunday, September 14
Homework assignment on Masteringphysics
Due before the lecture on Monday, September 15
Read Chapter 4 (p 104~126)
Read Chapter 5 (p 134~161)
Assignment
PHY 2048, Dr. Hebin Li
Goals for Chapter 3
To use vectors to represent the position of a body
To determine the velocity vector using the path of a body
To investigate the acceleration vector of a body
To describe the curved path of projectile
To investigate circular motion
To describe the velocity of a body as seen from different frames
of reference
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Describe motion in general
What determines the trajectory of a basketball and where the
basketball lands?
A satellite is flying in a circular orbit at constant speed, is it
accelerating?
How is the motion of a particle described by different moving
observers?
We need to extend the description of motion to two and three
dimensions.
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Going from 1D to 2D/3D
Position
1D 2D/3D
𝑥 𝒓
Time 𝑡 𝑡
Velocity 𝑣𝑥 𝒗
Acceleration 𝑎𝑥 𝒂
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Drawing vectors
• Draw a vector as a line with an arrowhead at its tip.
• The length of the line shows the vector’s magnitude.
• The direction of the line shows the vector’s direction.
• Figure 1.10 shows equal-magnitude vectors having the same
direction and opposite directions.
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• Two vectors may be added graphically using either the
parallelogram method or the head-to-tail method.
• Subtracting two vectors: reverse the direction of one
vector
• Adding three vectors: two at a time or adding all
vectors directly
Adding vectors graphically
Starting point
Ending point
Displacement
Order does not matter in vector additionThanks to Dr. Narayanan for the animation.
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Adding vectors graphically provides limited accuracy.
Vector components provide a general method for adding
vectors.
Any vector can be represented by an x-component Ax and
a y-component Ay.
Use trigonometry to find the components of a vector: Ax =
Acos θ and Ay = Asin θ, where θ is measured from the +x-
axis toward the +y-axis.
Components of a vector
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A unit vector has a magnitude of 1 with no units.
The unit vector 𝒊 points in the +x-direction, 𝒋 points in the +y-
direction, and 𝒌 points in the +z-direction.
Any vector can be expressed in terms of its components as
𝑨 = 𝐴𝑥 𝒊 + 𝐴𝑦 𝒋 + 𝐴𝑧 𝒌
This is equivalent to the expression: 𝑨 = (𝐴𝑥, 𝐴𝑦 , 𝐴𝑧)
Unit vectors
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𝑨 = 𝐴𝑥 𝒊 + 𝐴𝑦 𝒋 + 𝐴𝑧 𝒌
Given vectors 𝑨 and 𝑩
𝑩 = 𝐵𝑥 𝒊 + 𝐵𝑦 𝒋 + 𝐵𝑧 𝒌
The magnitude of 𝐴 |𝑨| = 𝐴𝑥2 + 𝐴𝑦
2 + 𝐴𝑧2
Adding two vectors 𝑨 + 𝑩 = (𝐴𝑥+𝐵𝑥) 𝒊 + (𝐴𝑦+𝐵𝑦) 𝒋 + (𝐴𝑧 + 𝐵𝑧) 𝒌
Multiplying a scalar 𝑐𝑨 = 𝑐𝐴𝑥 𝒊 + 𝑐𝐴𝑦 𝒋 + 𝑐𝐴𝑧 𝒌
Calculations with unit vectors and components
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Position vector
The position of an object in a 3D space is given by a position
vector. The vector has three components.
𝒓 = 𝑥 𝒊 + 𝑦 𝒋 + 𝑧 𝒌
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Average velocity
The average velocity between two points is the displacement
divide by the time interval between the two points. It has the
same direction as the displacement.
𝒗𝒂𝒗 =∆𝒓
∆𝑡
∆𝒓 = 𝒓2 − 𝒓1
In terms of components:
∆𝒓 = ∆𝑥 𝒊 + ∆𝑦 𝒋 + ∆𝑧 𝒌
𝑣𝑎𝑣−𝑥 =∆𝑥
∆𝑡
𝑣𝑎𝑣−𝑦 =∆𝑦
∆𝑡
𝑣𝑎𝑣−𝑧 =∆𝑧
∆𝑡
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Instantaneous velocity
The instantaneous velocity is the instantaneous rate of change
of position vector with respect to time. The instantaneous
velocity is always tangent to the path of motion.
𝒗 = lim∆𝑡→0
∆𝒓
∆𝑡=𝑑𝒓
𝑑𝑡
In terms of components:
𝑣𝑥 =𝑑𝑥
𝑑𝑡
𝑣𝑦 =𝑑𝑦
𝑑𝑡
𝑣𝑧 =𝑑𝑧
𝑑𝑡
PHY 2048, Dr. Hebin Li
Average acceleration
The average acceleration during a time interval ∆𝑡 is defined
as the velocity change divided by ∆𝑡.
𝒂𝒂𝒗 =∆𝒗
∆𝑡
In terms of components:
𝑎𝑎𝑣−𝑥 =∆𝑣𝑥∆𝑡
𝑎𝑎𝑣−𝑦 =∆𝑣𝑦∆𝑡
𝑎𝑎𝑣−𝑧 =∆𝑣𝑧∆𝑡
PHY 2048, Dr. Hebin Li
Instantaneous acceleration
The instantaneous acceleration is the
instantaneous rate of change of the
velocity with respect to time.
An object following a curved path is
accelerating, even if the speed is constant.
𝒂 = lim∆𝑡→0
∆𝒗
∆𝑡=𝑑𝒗
𝑑𝑡
In terms of components:
𝑎𝑥 =𝑑𝑣𝑥𝑑𝑡
=𝑑2𝑥
𝑑𝑡2
𝑎𝑦 =𝑑𝑣𝑦
𝑑𝑡=𝑑2𝑦
𝑑𝑡2
𝑎𝑧 =𝑑𝑣𝑧𝑑𝑡
=𝑑2𝑧
𝑑𝑡2
PHY 2048, Dr. Hebin Li
Parallel and perpendicular components of acceleration
The acceleration can be resolved into a component parallel to
the path and a component perpendicular to the path
The parallel component tells about changes in the speed
The perpendicular component tells about changes in the
direction of motion
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==
= =
= =
Summary: 1D -> 2D -> 3D
1D 2D 3D
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Projectile motion
A projectile is any body given an initial velocity that then
follows a path determined by the effects of gravity (for now,
we neglect the air resistance and the curvature and rotation of
the earth)
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The x and y motion of a projectile
Projectile motion can be
analyzed by separating x and y
motion
Horizontal direction (x)
Constant velocity, 𝑎𝑥 = 0𝑣𝑥 = 𝑣0𝑥
𝑥 = 𝑥0 + 𝑣0𝑥𝑡
Vertical direction (y)
Initial velocity, 𝑎𝑦 = −𝑔
𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡
𝑦 = 𝑦0 + 𝑣0𝑦𝑡 −1
2𝑔𝑡2
PHY 2048, Dr. Hebin Li
Analyzing projectile motion
x v0cos
0
t
y v0sin
0
t
12
gt2
vx v0cos
0
vy v0sin
0gt
These equations describe projectile
motion
Derivation :
The strategy is to consider a projectile motion in x and y
directions separately.
In x direction:
Initial x-component velocity: 𝑣0𝑥 = 𝑣0 cos 𝛼0x-component acceleration: 𝑎𝑥 = 0Thus, 𝑣𝑥 = 𝑣0 cos 𝛼0 and 𝑥 = 𝑣0𝑥𝑡 = (𝑣0cos 𝛼0)𝑡
In y direction:
Initial y-component velocity: 𝑣0𝑦 = 𝑣0 sin 𝛼0y-component acceleration: 𝑎𝑦 = −𝑔
Thus,
𝑣𝑦 = 𝑣0𝑦 + 𝑎𝑦𝑡 = 𝑣0 sin 𝛼0 − 𝑔𝑡;
𝑦 = 𝑣0𝑦𝑡 +1
2𝑎𝑦𝑡
2 = (𝑣0 sin 𝛼0)𝑡 −1
2𝑔𝑡2.
PHY 2048, Dr. Hebin Li
Trajectory of projectile motion
x v0cos
0
t
y v0sin
0
t
12
gt2
vx v0cos
0
vy v0sin
0gt
An equation of the trajectory in terms
of x and y by eliminating t.
𝑦 = (tan𝛼0)𝑥 −𝑔
2𝑣02cos2𝛼0
𝑥2
The trajectory is parabolic
Derivation:
𝑡 =𝑥
𝑣0 cos 𝛼0
𝑦 = 𝑣0 sin 𝛼0𝑥
𝑣0 cos 𝛼0−1
2𝑔
𝑥
𝑣0 cos 𝛼0
2
= tan𝛼0 𝑥 −𝑔
2𝑣02𝑐𝑜𝑠2𝛼0
𝑥2
(1)
(2)
From (1), we got
Plug it into (2), we have
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The effects of air resistance
• Calculations become
more complicated.
• Acceleration is not
constant.
• Effects can be very
large.
• Maximum height and
range decrease.
• Trajectory is no longer a
parabola.
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Circular motion & uniform circular motion
𝒗 𝒗 𝒗
𝒂
𝒂
𝒂
For uniform circular motion, the speed is constant and the
acceleration is perpendicular to the velocity
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Centripetal acceleration
Comparing the two triangles in (a) and (b),
|∆ 𝑣|
𝑣1=∆𝑠
𝑅
=> |∆ 𝑣| =𝑣1∆𝑠
𝑅
𝑎 = lim∆𝑡→0
|∆ 𝑣|
∆𝑡= lim
∆𝑡→0
𝑣1𝑅
∆𝑠
∆𝑡=𝑣2
𝑅
with 𝑣1 = 𝑣2 = 𝑣
Centripetal acceleration
Magnitude: or𝑎𝑟𝑎𝑑 =𝑣2
𝑅
The direction is perpendicular to the
velocity and inward along the radius. It is
changing.
𝑎𝑟𝑎𝑑 =4𝜋2𝑅
𝑇2
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Nonuniform circular motion
If the speed varies, the motion is
nonuniform circular motion.
The acceleration has both the radial
and tangential components.
𝑎𝑟𝑎𝑑 =𝑣2
𝑅𝑎𝑡𝑎𝑛 =
𝑑| 𝑣|
𝑑𝑡
PHY 2048, Dr. Hebin Li
Relative velocity
The velocity of a moving body seen by a particular observer
is called the velocity relative to that observer, or simply the
relative velocity.
A frame of reference is a coordinate system plus a time scale.
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Relative velocity in one dimension
The position of point P relative to
frame B is 𝑥𝑃/𝐵, the position of the
origin of frame B relative to frame A is
𝑥𝐵/𝐴, then the position of point P
relative to frame A is given by
𝑥𝑃/𝐴 = 𝑥𝑃/𝐵 + 𝑥𝐵/𝐴
If P is moving relative to frame B and
frame B is moving relative to frame A,
then the x-velocity of P relative to
frame A is
𝑣𝑃/𝐴−𝑥 = 𝑣𝑃/𝐵−𝑥+𝑣𝐵/𝐴−𝑥
PHY 2048, Dr. Hebin Li
Relative velocity in 2/3 dimensions
The concept of relative velocity can be extended into 2/3 dimensions by
using vectors
𝑟𝑃/𝐴 = 𝑟𝑃/𝐵 + 𝑟𝐵/𝐴
𝑣𝑃/𝐴 = 𝑣𝑃/𝐵 + 𝑣𝐵/𝐴