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

DDxx = horizontal range = horizontal range

DDyy = initial height = initial height

above groundabove ground

t = total time in the airt = total time in the air““Hang Time”Hang Time”

IMPORTANT FACTSIMPORTANT FACTSThere 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

DDxx = V = Vxxttvertical

D = 0.5gtD = 0.5gt22

Object Launched at an AngleObject Launched at an AngleV = initial velocityV = initial velocity = launch angle= launch angle DDyy = maximum height = maximum height

t = Hang Time = total time in airt = Hang Time = total time in air DDxx = 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

DDxx = v = vxxtt

verticalverticalVVyy = V sin = V sin

DDyy = 0.5(0.5)V = 0.5(0.5)Vyyttt = t = 2V2Vyy/g/g

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

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Learn more about Learn more about projectile motionprojectile motion

at these links:at these links:

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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

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..

The speed v is theThe speed v is thecircumference 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

G Forces are sometimes confused with pulling G’s. G’s are multiples of

the acceleration of gravity.

Puling 4 G’s really means having a centripetal acceleration of

4 x 9.8 = 39.2 m/s/s

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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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:

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View pendulum simulations at:View pendulum simulations at:

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