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

Center, copy your notes there at no cost, and leave them there for the student to

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.

http://faculty.fiu.edu/~hebli/teaching/phy-2048-schedule/


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


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


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.


Going from 1D to 2D/3D

Position

1D 2D/3D

𝑥 𝒓

Time 𝑡 𝑡

Velocity 𝑣𝑥 𝒗

Acceleration 𝑎𝑥 𝒂


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.


• 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.


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


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


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


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


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


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


𝑣𝑥 =𝑑𝑥

𝑑𝑡

𝑣𝑦 =𝑑𝑦

𝑑𝑡

𝑣𝑧 =𝑑𝑧

𝑑𝑡


Average acceleration

The average acceleration during a time interval ∆𝑡 is defined

as the velocity change divided by ∆𝑡.

𝒂𝒂𝒗 =∆𝒗

∆𝑡


𝑎𝑎𝑣−𝑥 =∆𝑣𝑥∆𝑡

𝑎𝑎𝑣−𝑦 =∆𝑣𝑦∆𝑡

𝑎𝑎𝑣−𝑧 =∆𝑣𝑧∆𝑡


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


𝑎𝑥 =𝑑𝑣𝑥𝑑𝑡

=𝑑2𝑥

𝑑𝑡2

𝑎𝑦 =𝑑𝑣𝑦

𝑑𝑡=𝑑2𝑦

𝑑𝑡2

𝑎𝑧 =𝑑𝑣𝑧𝑑𝑡

=𝑑2𝑧

𝑑𝑡2


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


==

= =

= =

Summary: 1D -> 2D -> 3D

1D 2D 3D


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)


PhET simulation

http://phet.colorado.edu/



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


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.


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


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.


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


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


PhET simulation




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


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.


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


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

chapter 3 motion in two or three dimensions

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