ch04 lecture motion in two and three...

Chapter 4

Motion in Two and Three

Dimesions

4 Motion in Two and Three Dimensions

25 October 2018 2 PHY101 Physics I © Dr.Cem Özdoğan


4-2 Position and Displacement



Example: Two-dimensional motion (rabbit position)


4-3 Average Velocity and Instantaneous Velocity

• If a particle moves through a displacement of Dr in Dt time, then the average velocity is:

• In the limit that the Dt time shrinks to a single point in time, the average velocity is approaches instantaneous velocity.

• This velocity is the derivative of displacement with respect to time.

Example: A particle moves through displacement (12 m)i + (3.0 m)k in 2.0 s:


Visualize displacement and instantaneous velocity:

Note: a velocity vector does not extend from one point to another, only

shows direction and magnitude.



Example: Two-dimensional motion (rabbit velocity)



4-4 Average and Instantaneous Accelerations

• If we shrink Dt to zero, then the average acceleration value

approaches to the instant acceleration value, which is the derivative of

velocity with respect to time:

• Following the same definition as in average velocity,


Example: Two-dimensional motion

(rabbit run)



4-5 Projectile Motion

• Projectile; a particle moves in a vertical plane with some initial

velocity but its acceleration is always the free-fall acceleration which

is downward.

• This particle’s motion is called projectile motion.

• Thrown ball

• Bullet (ballistics considered as projectile motion)

• Dropped package

! IMPORTANT NOTE !

We have assumed that air through which

the projectile moves has no effect on its

motion.

Launched with an initial velocity v0


A stroboscopic photograph of two golf

balls.

The initial velocity of the projectile is:

0xa

gay

Here,

Therefore we can decompose two-dimensional motion

into 2 one-dimensional problems



4-6 Projectile Motion Analyzed

The projectile's trajectory. Its path through space (traces a parabola)


Horizontal Range (assuming no external forces)

The horizontal range of a projectile is the horizontal distance when it

returns to its launching height.

The distance equations in the x- and y- directions respectively:

Eliminating t;




AK Lecture Notes


The Effects of the Air



Example: Projectile dropped from airplane



Example: Cannonball to pirate ship



Exercise:

• A policeman chases a thief across city rooftops. They are both running at 5 m/s when they come to a gap between buildings that is 4 m wide and has a drop of 3 m.

• The thief leaps at 5 m/s at an angle of 45°. Does he clear the gap?

• The policeman leaps at 5 m/s horizontally. Does he clear the gap? 0x

0y



4-7 Uniform Circular Motion


• As the direction of the velocity of the particle changes, there is an acceleration!!!

CENTRIPETAL (center-seeking)

ACCELERATION

! IMPORTANT NOTE !

When the motion is non-uniform circular

motion, the speed and the direction change.

Velocity and acceleration have:

Constant magnitude

Changing direction



Acceleration is called centripetal acceleration

Means “center seeking”

Directed radially inward

The period of revolution is:

The time it takes for the particle go around the closed path

exactly once



Example: Top gun pilots

We assume the turn is made with uniform circular

motion.

Then the pilot’s acceleration is centripetal and has

magnitude a given by a =v2/R.

Also, the time required to complete a full circle

is the period given by T =2pR/v

Because we do not know radius R, let’s solve for R

from the period equation for R and substitute into

the acceleration eqn.

Speed v here is the (constant) magnitude of the

velocity during the turning.

To find the period T of the motion, first note that

the final velocity is the reverse of the initial

velocity. This means the aircraft leaves on the

opposite side of the circle from the initial point

and must have completed half a circle in the given

24.0 s. Thus a full circle would have taken T 48.0

s.

Substituting these values into our equation for a,

we find



4-8 Relative motion in one-dimension (1-D)

‘The velocity of a particle depends on the reference frame

of whoever is observing the velocity.’

Measures of position and velocity depend on the reference frame of

the measurer

How is the observer moving?

Our usual reference frame is

that of the ground


• Suppose that they both measure the position

of the car at a given moment. Then:

• Positions in different frames are related by:

• Taking the derivative, we see velocities are

related by:

But accelerations (for non-accelerating

reference frames, aBA

= 0) are related by

Read subscripts “PA”, “PB”, and

“BA” as “P as measured by A”,

“P as measured by B”, and “B as

measured by A”.

Frames A and B are each watching

the movement of object P



Example:

Frame A: x = 2 m, v = 4 m/s

Frame B: x = 3 m, v = -2 m/s

P as measured by A: xPA

= 5 m, vPA

= 2 m/s, a = 1 m/s2

So P as measured by B:

xPB

= xPA

+ xAB

= 5 m + (2m – 3m) = 4 m

vPB

= vPA

+ vAB

= 2 m/s + (4 m/s – -2m/s) = 8 m/s

a = 1 m/s2



Example: Relative motion, 1-D



4-9 Relative motion in two-dimensions (2-D)

• The same as in one dimension, but now with vectors:

• A and B, the two observers, are watching P, the moving particle, from

their origins of reference.

• B moves at a constant velocity with respect to A, while the

corresponding axes of the two frames remain parallel.

• rPA refers to the position of P as observed by A, and so on.

• Again, observers in different frames will see the same acceleration.


Calculate the wind velocity for the

situation shown in Figure above. The

plane is known to be moving at 45.0

m/s due north relative to the air mass,

while its velocity relative to the ground

(its total velocity) is 38.0 m/s in a

direction 20.0o west of north.

(16 m/s, 35.6o)

Calculate the magnitude and direction

of the boat’s velocity relative to an observer

on the shore, vtot. The velocity of the boat,

vboat , is 0.75 m/s in the y-direction relative

to the river and the velocity of the river,

vriver , is 1.20 m/s to the right.

(1.42 m/s, 32o)



Example: Relative motion, 2-D



Example: (OpenStax Example 3.4) During a fireworks display, a shell is shot into the air with an

initial speed of 70.0 m/s at an angle of 75.0o above the horizontal, as illustrated in

Figure. The fuse is timed to ignite the shell just as it reaches its highest point above the

ground.

(a) Calculate the height at which the shell explodes.

(b) How much time passed between the launch of the shell and the explosion?

(c) What is the horizontal displacement of the shell when it explodes?


(OpenStax Example 3.5) Suppose a large

rock is ejected from the volcano with a speed

of 25.0 m/s and at an angle 35.0o above the

horizontal, as shown in Figure. The rock

strikes the side of the volcano at an altitude

20.0 m lower than its starting point.

(a) Calculate the time it takes the rock to follow this path.

(b) What are the magnitude and direction of the rock’s velocity at impact?

Example:


(Serway 4.6) What is the centripetal acceleration of the Earth as it moves in its orbit

around the Sun?

Example:


(OpenStax Example 3.8) An airline passenger drops a coin while

the plane is moving at 260 m/s. What is the velocity of the coin

when it strikes the floor 1.50 m below its point of release: (a)

Measured relative to the plane? (b) Measured relative to the

Earth?

Example:


4 Solved Problems

1. A particle leaves the origin with an initial velocity v = (3.00i) m/s and a constant

acceleration a = (-1.00i-0.500j) m/s2. When it reaches its maximum x coordinate,

what are its (a) velocity and (b) position vector?


2. The current world‐record motorcycle jump is 77.0 m. Assume that he left the

take‐off ramp at 12.0° to the horizontal and that the take‐off and landing heights

are the same. Neglecting air drag, determine his take‐off speed.

4 Solved Problems


3. In Figure, a stone is projected at a cliff of height with an initial speed of 42.0 m/s

directed at angle 60° above the horizontal. The stone strikes at 5.50 s after

launching. Find (a) the height of the cliff (h), (b) the speed of the stone just before

impact at , and (c) the maximum height (H) reached above the ground.

4 Solved Problems


3. (Continued) In Figure, a stone is projected at a cliff of height with an initial speed

of 42.0 m/s directed at angle 60° above the horizontal. The stone strikes at 5.50 s

after launching. Find (a) the height of the cliff (h), (b) the speed of the stone just

before impact at , and (c) the maximum height (H) reached above the ground..

4 Solved Problems


4. (Serway Example 4.2) A long jumper leaves the ground at an angle of 20.0° above

the horizontal and at a speed of 11.0 m/s. (a) How far does he jump in the

horizontal direction? (b) What is the maximum height reached?

4 Solved Problems


5. (Serway Example 4.5) A ski jumper

leaves the ski track moving in the

horizontal direction with a speed of 25.0

m/s as shown in Figure. The landing

incline below her falls off with a slope

of 35.0°. Where does she land on the

incline?

4 Solved Problems


6. An Earth satellite moves in a circular orbit 640 km above Earth's surface with a

period of 98.0 min. What are the (a) speed and (b) magnitude of the centripetal

acceleration of the satellite?

4 Solved Problems


7. A car exhibits a constant acceleration of

0.300 m/s2 parallel to the roadway. The car

passes over a rise in the roadway such that

the top of the rise is shaped like a circle of

radius 500 m. At the moment the car is at

the top of the rise, its velocity vector is

horizontal and has a magnitude of 6.00 m/s.

What are the magnitude and direction of the

total acceleration vector for the car at this

Instant?

4 Solved Problems


Position Vector

Locates a particle in 3-space

Displacement

Change in position vector

Average and Instantaneous Accel.

Average and Instantaneous Velocity

Eq. (4-2)

Eq. (4-8) Eq. (4-15)

4 Summary

Eq. (4-1)

Eq. (4-3)

Eq. (4-4)

Eq. (4-10) Eq. (4-16)


Projectile Motion

Flight of particle subject only to

free-fall acceleration (g)

Trajectory is parabolic path

Horizontal range:

Uniform Circular Motion

Magnitude of acceleration:

Time to complete a circle:

Relative Motion

For non-accelerating reference

frames

Eq. (4-34)

Eq. (4-44)

Eq. (4-22)

Eq. (4-23)

Eq. (4-25)

Eq. (4-26)

Eq. (4-35)

Eq. (4-45)

4 Summary


Additional Materials

4 Motion in Two and Three Dimensions


Centripetal acceleration, proof of a = v2/r

cos,sin

,

vvvv

vdt

dxv

dt

dy

yx

x

P

y

P

Note




4.01 Draw two-dimensional

and three-dimensional

position vectors for a

particle, indicating the

components along the axes

of a coordinate system.

4.02 On a coordinate system,

determine the direction and

magnitude of a particle's

position vector from its

components, and vice versa.

4.03 Apply the relationship

between a particle's

displacement vector and its

initial and final position

vectors.

Learning Objectives



4.04 Identify that velocity is a

vector quantity and thus has

both magnitude and

direction and also has

components.



velocity vectors for a


components along the axes

of the coordinate system.

4.06 In magnitude-angle and

unit-vector notations, relate

a particle's initial and final

position vectors, the time

interval between those

positions, and the particle’s

average velocity vector.

4.07 Given a particle’s position

vector as a function of time,

determine its

(instantaneous) velocity

vector.

Learning Objectives


4-4 Average Acceleration and Instantaneous Acceleration

4.08 Identify that acceleration

is a vector quantity, and thus

has both magnitude and

direction.



acceleration vectors for a


components.

4.10 Given the initial and final

velocity vectors of a particle

and the time interval,

determine the average

acceleration vector.

4.11 Given a particle's velocity

vector as a function of time,

determine its

(instantaneous) acceleration

vector.

4.12 For each dimension of

motion, apply the constant-

acceleration equations

(Chapter 2) to relate

acceleration, velocity,

position, and time.

Learning Objectives


4-5_6 Projectile Motion

4.13 On a sketch of the path

taken in projectile motion,

explain the magnitudes and

directions of the velocity and

acceleration components

during the flight.

4.14 Given the launch velocity

in either magnitude-angle or

unit-vector notation,

calculate the particle's

position, displacement, and

velocity at a given instant

during the flight.

4.15 Given data for an instant

during the flight, calculate

the launch velocity.

Learning Objectives



4.16 Sketch the path taken in

uniform circular motion and

explain the velocity and

acceleration vectors

(magnitude and direction)

during the motion.

4.17 Apply the relationships

between the radius of the

circular path, the period, the

particle's speed, and the

particle's acceleration

magnitude.

Learning Objectives


4-8 Relative Motion in One Dimension

4.18 Apply the relationship between a particle's position,

velocity, and acceleration as measured from two reference

frames that move relative to each other at a constant velocity

and along a single axis.

Learning Objectives


4-9 Relative Motion in Two Dimensions

4.19 Apply the relationship between a particle's position,

velocity, and acceleration as measured from two reference

frames that move relative to each other at a constant velocity

and in two dimensions.

Learning Objectives

ch04 lecture motion in two and three...

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