Four Major Types ofFour Major Types ofTwo Dimensional MotionTwo Dimensional Motion

1. 1. Projectile MotionProjectile Motion

2. 2. Circular MotionCircular Motion

3. 3. Rotational MotionRotational Motion

4. 4. Periodic MotionPeriodic Motion

Projectile motion problems are best solved byProjectile motion problems are best solved bytreating horizontal and vertical motion separately.treating horizontal and vertical motion separately.

IMPORTANTIMPORTANTGravity only affects vertical motion.Gravity only affects vertical motion.

There are two general typesThere are two general typesof projectile motion situations.of projectile motion situations.

1. object launched horizontallyobject launched horizontally2. object launched at an angleobject launched at an angle

Object Launched HorizontallyObject Launched Horizontallyvvxx = initial horizontal velocity = initial horizontal velocity

RRxx = horizontal range = horizontal range

h = initial height h = initial height above groundabove ground

t = total time in the airt = total time in the airIMPORTANT FACTSIMPORTANT FACTS

There is no horizontal acceleration.There is no horizontal acceleration.There is no initial vertical velocity.There is no initial vertical velocity.The horizontal velocity is constant.The horizontal velocity is constant.

Time is the same for both vertical and horizontal.Time is the same for both vertical and horizontal.

horizontal

RRxx = v = vxxttvertical

h = 0.5gth = 0.5gt22

Object Launched at an AngleObject Launched at an Anglev = initial velocityv = initial velocity = launch angle= launch angle h = maximum heighth = maximum height

t = total time in airt = total time in air RRxx = horizontal range = horizontal range

IMPORTANT FACTSIMPORTANT FACTSThe horizontal velocity is constant.The horizontal velocity is constant.

It rises and falls in equal time intervals.It rises and falls in equal time intervals.It reaches maximum height in half the total time.It reaches maximum height in half the total time.

Gravity only effects the vertical motion.Gravity only effects the vertical motion.

horizontalhorizontalvvxx = v cos = v cos

RRxx = v = vxxtt

verticalverticalvvyy = v sin = v sinh = vh = vyyt/4t/4t = t = 2v2vyy/g/g

Learn more about projectile motionLearn more about projectile motionat these links:at these links:

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View projectile motion simulations at:View projectile motion simulations at:

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• constant initial velocity versus how the horizontal constant initial velocity versus how the horizontal range changes with angle; plot “range vs angle”range changes with angle; plot “range vs angle”

• constant initial velocity versus how total time in air constant initial velocity versus how total time in air changes with angle; plot “total time vs angle”changes with angle; plot “total time vs angle”

• constant initial velocity versus how maximum height constant initial velocity versus how maximum height changes with angle; plot “height vs angle”changes with angle; plot “height vs angle”

• constant angle versus how the horizontal range constant angle versus how the horizontal range changes with initial velocity; plot “range vs velocity”changes with initial velocity; plot “range vs velocity”

• constant angle versus how the total time in the air constant angle versus how the total time in the air changes with initial velocity; plot “time vs velocity”changes with initial velocity; plot “time vs velocity”

• constant angle versus how the maximum height constant angle versus how the maximum height changes with initial velocity; plot “height vs velocity”changes with initial velocity; plot “height vs velocity”

Suggested Constructivist ActivitiesSuggested Constructivist Activities

Students useStudents use simulations tosimulations to complete data tablescomplete data tablesand make graphs of the following situations:and make graphs of the following situations:

object moves inobject moves incircular pathcircular path

about an about an point point((“revolves”“revolves”))

According to Newton’s According to Newton’s First Law of MotionFirst Law of Motion,,

objects move in a straight line unless a forceobjects move in a straight line unless a force

makes them turn. An external force is necessary makes them turn. An external force is necessary

to make an object follow a circular path.to make an object follow a circular path.

This force is called aThis force is called a

CENTRIPETALCENTRIPETAL ( (““center seekingcenter seeking”) ”) FORCEFORCE..

Since Since everyevery unbalanced force causes an object unbalanced force causes an object

to accelerate in the direction of that forceto accelerate in the direction of that force

((Newton’s Second LawNewton’s Second Law), a centripetal force), a centripetal force

causes a causes a CENTRIPETAL ACCELERATIONCENTRIPETAL ACCELERATION. This. This

acceleration results from a change in direction,acceleration results from a change in direction,

and and does not imply a change in speeddoes not imply a change in speed,,

although speed may also change. although speed may also change.

Centripetal force and acceleration may be caused by:Centripetal force and acceleration may be caused by:•gravitygravity - planets orbiting the sun - planets orbiting the sun

•frictionfriction - car rounding a curve - car rounding a curve•a rope or corda rope or cord - swinging a mass on a string - swinging a mass on a string

r m

In all cases, a In all cases, a mass mmass m moves in a circular path moves in a circular pathof of radius rradius r with a with a linear speed vlinear speed v. The time to make. The time to makeone complete revolution is known as the one complete revolution is known as the periodperiod, , TT..

vThe speed v is theThe speed v is the

circumference divided by the period.circumference divided by the period.

v = 2v = 2r/Tr/T

The formula for The formula for centripetal accelerationcentripetal acceleration is: is:

aacc == v v22//rrand and centripetal forcecentripetal force is: is:

FFcc == mmaacc == mvv22//rrm = mass in kgm = mass in kg

v = linear velocity in m/sv = linear velocity in m/sFFcc = centripetal force in N = centripetal force in N

r = radius of curvature in mr = radius of curvature in maacc = centripetal acceleration in m/s = centripetal acceleration in m/s22

Learn more about circular motionLearn more about circular motionat these links:at these links:

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View circular motion simulations at:View circular motion simulations at:

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object moves inobject moves incircular path aboutcircular path about

an an point pointor axisor axis

((“rotates”“rotates” or or “spins”“spins”))

The amount that an object rotates is itsThe amount that an object rotates is itsangular displacementangular displacement..

angular displacementangular displacement, , , is given in, is given in

degreesdegrees, , radiansradians, or , or rotationsrotations..

1 rotation1 rotation = = 360 deg360 deg = = 22 radians radians

The The time rate change of an object’stime rate change of an object’sangular displacementangular displacement is itsis its

angular velocityangular velocity..

angular velocityangular velocity,, , is given in, is given indeg/sdeg/s, , rad/srad/s, , rpmrpm,, etc...etc...

The The time rate change of an object’stime rate change of an object’sangular velocityangular velocity is its is itsangular accelerationangular acceleration..

Angular accelerationAngular acceleration, , , is given in, is given indeg/sdeg/s22, , rad/srad/s22, , rpm/srpm/s,, etc... etc...

Formulas for rotational motion follow anFormulas for rotational motion follow anexact parallel with linear motion formulas.exact parallel with linear motion formulas.The only difference is a change in variablesThe only difference is a change in variables

and a slight change in their meanings.and a slight change in their meanings.

ConstantConstant

LINEARLINEAR

vvff = v = vii + at + atd = vd = vavavtt

vvavav = (v = (vff + v + vii)/2)/2

d = vd = viit + 0.5att + 0.5at22

vvff22 = v = vii

22 + 2ad + 2ad

ROTATIONALROTATIONAL

ff= = ii + +tt==avavtt

vvavav = =((ff++ii)/2)/2

==iitt++0.50.5tt22

ff22= = ii

22++22

PPEERROODDIICC

MOTIONMOTION

any motion in whichany motion in whichthe path of the objectthe path of the objectrepeats itself in equalrepeats itself in equal

time intervalstime intervals

The The simple pendulumsimple pendulumis a great example ofis a great example ofthis type of motion.this type of motion.

The The periodperiod, , TT, of a simple pendulum, of a simple pendulum((time needed for one complete cycletime needed for one complete cycle))

is approximated by the equation:is approximated by the equation:

T 2lg

T 2lg

where where l is the length of the penduluml is the length of the pendulumand and g is the acceleration of gravityg is the acceleration of gravity..

Learn more about pendulums andLearn more about pendulums andperiodic motion at these links:periodic motion at these links:

link1, link2, link3, link4, link5 link1, link2, link3, link4, link5

View pendulum simulations at:View pendulum simulations at:

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