03 Lecture Lam freedman physics 13ed

8/10/2019 03 Lecture Lam freedman physics 13ed

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Copyright 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint Lectures forUniversity Physics, Twelfth Edition

Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 3

Motion in Two orThree Dimensions

Modified by Pui Lam 5_29_2013


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Goals for Chapter 3 To recognize that in 2 or 3-dimensions that the

velocity vector and the acceleration vector need notbe parallel.

Examples

projectile motion uniform and non-uniform circular motion general curve motion Distinguish constant speed vs constant velocity Use dot product between v and a vectors to

determine whether an object is speeding up orslowing down

To investigate relative velocity


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Position relative to the originFigure 3.1 For general motion in 3-dimension, the position vector

relative to the origin can have components in x , y , and z directions. The path of a particle is generally a curve.

!

r (t ) = x(t ) i + y(t ) j + z(t ) k


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Average velocity and Instantaneous velocityFigure 3.2 The average velocity between two points will have the

same direction as the displacement

** Instantaneous velocity=!

v !d

!

rdt

=dxdt

i +dydt

j +dzdt

k

** Instantaneous velocity is a vector tangent to the path


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Example - Projectile motion A projectile is any body given an initial velocity that follows a

path determined by the effects of gravity and air resistance. If air resistance is negligible, then projectile motion = motion

under constant acceleration, a x =0 and a y= -g


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Equations for Projectile Motion (neglect air-resistance)

Find x(t) and y(t) for a projectile motion given its initial positionand initial velocity.

Calculate the subsequent velocity as a function of time.


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Projectile motion - numerical example IQ . Given : v o = 10 m / s , ! o = 30

o , and

g ~ 10 m/s 2 . (Assume no air resistance)

(1) Find the time it reaches the top.

(2) Find the velocity vector at the top

(3) Find the range.

(4) Find the velocity vector when it returns to the ground


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Effect of air resistance

Q . Sketch the path if there is air - resistance.


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Projectile motion - numerical example II

Q . A basket player is shooting a baskeball from a height of 2m.The basket ball rim is 5 m away (horizontal distance) and 3 m high.

He must make the basket in 1 sceond.

At what angle ( ! ) from the horizontal and at what initial speed

must he throw the ball to make the basket in 1 second?

(Assume no air resistance and g=10m/s 2 )


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The instantaneous acceleration vectorFigure 3.6 The acceleration vector is non-zero as long as there is a change in

the velocity vector. The change can be either the magnitude OR the direction of the

velocity, OR both.

In 2- and 3-dimension, acceleration vector needs not be in the

same direction of velocity vector. Example: Car going around acurve.


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Uniform circular motion and centripetal acceleration For circular motion:

x(t)=rcos!

(t), y(t)=rsin!

(t)For uniform circular motion:! increases uniformly " ! (t)= # t" x(t)=rcos # t, y(t)=rsin # t

(e.g. 10 rpm " # = 10(2 $ rad . ) /60 s )

From these, one can calc;!

v(t) and!

a(t)

(You fill in the details)

Result:

!

a c =

!

v2

r ; !

v =

magnitude of velocity=speed

Note: In uniform circular motion, the

speed

is unchanged,but the velocity vector changes => there is an acceleration.

The acceleration is called centripetal acceleration pointing to the center.


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Example of non-uniform circular motion

Circular motion

doesn t mean a completecircle, could be part of acircle or any curve.

In non-uniform circularmotion, the speed is NOTconstant, hence there isalso a tangential

acceleration , in additionto the centripetalacceleration.


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In daily language, acceleration means speeding up and deceleration meansslowing down.

In Physics, acceleration (vector) = change of velocity (vector) wrt time.

An object has a non-zero acceleration (vector) whenever there is a change invelocity (vector); the object can be speeding up, slowing down, or keeping thesame speed.

!

v !

a > 0 ! speedingup !

v !

a < 0 ! slowing down !

v !

a = 0 ! constant speed

Acceleration daily usage vs. physics usage


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Relative velocity on a straight road

What is the velocity vector of the Truck with respect to you?Let

!

VT/E = ! 20 j;!

VY/E = + 30 j!

VT/Y =!

VT/E +!

VE/Y"

!

VT/Y =!

VT/E !!

VY/E= ! 20 ! 30 = ! 50 j

Same formula applies for2- or 3-dimensiosn.

Example :!

VT/E = ! 20 j ,!

VY/E = 30 j


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Question: The compass of an airplane indicates that it is

headed north and the plane is moving at 240km/hr throughthe air. If there is wind of 100km/hr from west to east, whatis the velocity of the plane relative to the ground?

03 Lecture Lam freedman physics 13ed

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