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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Thursday, January 24, 2008.

Thor's Emerald Helmet. Credit & Copyright: Robert Gendler

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Thursday, January 24, 2008.

• Topics:

• Updated Course Information

• Review of motion in one dimension

• Motion in two dimensions:

• Vectors

• Position, velocity, and acceleration in two and three dimensions

• Projectile motion

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121Updated course information

• The Physics 121 workshops will start on Monday January28.

• The physics 121 laboratories will start also on MondayJanuary 28.

• There will be no lecture on Thursday 1/31. I will be inEurope from Wednesday 1/30 until Monday 2/4.

• Anyone who did not take the Diagnostic Test on Tuesday1/22 needs to make up this test on Thursday morning 1/31 at9.40 am in Hoyt (it will take 45 minutes to complete thisDiagnostic Test).

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121 homework set # 1.Due 2/2/08.

Weighted meanof v/t.

Use spread sheets!

Error propagation.

Relative velocity.

Integrate v to find d.

Relative velocity.

Complicated projectilemotion.

Relative velocity.

Obtaining a from x(t).

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Review of motion in one dimension.

• Translational motion on onedimension can be described interms of three parameters:

• The position x(t): units m.

• The velocity v(t): units m/s.

• The acceleration a(t): units m/s2.

• An important special case is thecase of constant acceleration(acceleration independent oftime)

a =

dv

dt= constant

v(t) = v0

+ a t

x(t) = x

0+ v

0t +

1

2a t

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Review of motion in one dimension.

a =

dv

dt= - g

v(t) = v0 - g t

y(t) = y

0+ v

0t -

1

2g t

2

Note: in this format, g is assumed to be the magnitude of the gravitational acceleration.

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two or three dimensions.Vectors.

• In order to study motion in twodimensions, we need to introducethe concepts of vectors.

• Position, velocity, andacceleration in two- or three-dimensions are determined by notonly specifying their magnitude,but also their direction.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Two-dimensional motion.

• The same displacement can beachieved in many different ways.

• Instead of specifying a headingand distance that takes you fromthe origin of your coordinatesystem to your destination, youcould also indicate how many kmNorth you need to travel and howmany km East.

• In either case you need to specifytwo numbers and this type ofmotion is called two dimensionalmotion.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Vector manipulations.

• Any complicated type of motioncan be broken down into a seriesof small steps, each of which canbe specified by a vector.

• I will make the assumption thatyou have read the details aboutvector manipulations:

• Vector addition

• Vector subtraction

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Vector components.

• Although we can manipulatevectors using various graphicaltechniques, in most cases theeasiest approach is to decomposethe vector into its componentsalong the axes of the coordinatesystem you have chosen.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Vector components.

• Using vector components, vectoraddition becomes equivalent toadding the components of theoriginal vectors.

• The sum of the x and ycomponents can be used toreconstruct the sum vector.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Other vector manipulations:the scalar product.

• The scalar product (or dotproduct) between two vectors is ascalar which is related to themagnitude of the vectors and theangle between them.

• It is defined as:!

a

b

!ai

!b =!a!b cos!

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Other vector manipulations:the scalar product.

• In terms of the components of a and b, the scalar product isequal to

• Usually, you will use the component form to calculate thescalar product and then use the vector form to determine theangle between vectors a and b.

!ai

!b = a

xb

xii i + a

yb

yji j+ a

zb

zkik +

axb

y+ a

yb

x( ) ii j+ axb

z+ a

zb

x( ) iik + ayb

z+ a

zb

y( ) jik

= axb

x+ a

yb

y+ a

zb

z

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Other vector manipulations:the vector product.

• The vector product between twovectors is a vector whosemagnitude is related to themagnitude of the vectors and theangle between them, and whosedirection is perpendicular to theplane defined by the vectors.

• The vector product is defined as

!c =!a!b sin!

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Other vector manipulations:the vector product.

• Usually the vector product is calculated by using thecomponents of the vectors a and b:

!a !!b =

i j k

ax ay az

bx by bz

=

= aybz " azby( ) i + azbx " axbz( ) j + axby " aybx( ) k

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Defining motion in two dimensions.

One Dimension

• Position x• Displacement Δx• Velocity: displacement per unit

time. Sign is equal to the sign ofthe displacement Δx

• Acceleration: change in velocityΔv per unit time. Sign is equal tothe sign of the velocity differenceΔv.

Two/Three Dimensions

• Position vector r• Displacement vector Δr• Velocity vector: change in the

position vector per unit time.The direction is equal to thedirection of the displacementvector Δr.

• Acceleration vector: change inthe velocity vector per unit time.The direction is equal to thedirection of the velocitydifference vector Δv.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions.

• The direction of the velocityvector is tangent to the path ofthe object.

• The direction of the accelerationvector is more complicated and ingeneral is not pointing in thesame direction as the velocity.

• In non-zero acceleration in two orthree dimensions does not need toresult in a change of the speed ofan object. It may only change thedirection of motion and not itsmagnitude (circular motion).

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions.

• When an object moves in twodimensions, we can consider twocomponents of its motionseparately.

• For example, in the case ofprojectile motion, thegravitational acceleration onlyinfluences the motion in thevertical direction.

• In the absence of an externalforce, there is no acceleration inthe horizontal direction, and thevelocity in that direction is thusconstant.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions.

• Let’s practice what we just discussed and focus our attentionon problem Q2.11.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions:projectile motion.

• To study projectile motion wedecompose the motion into itstwo components:

• Vertical motion:• Defines how long it will take for

the projectile to hit the ground

• Horizontal motion:• During this time interval, the

distance traveled by the projectileis

t =2v

0sin!

0

g

x = v0cos!

0( )2v

0sin!

0

g

"#$

%&'=v0

2

gsin2!

0= R

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Motion in two dimensions:projectile motion.

• Let’s practice what we just discussed and focus our attentionon problem Q2.12.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Done! See you next week on Tuesday.Please review Chapter 4.

Adirondack Rock on Mars Credit: Mars Exploration Rover Mission, JPL, NASA