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show mode.show mode.

Warm-up – Shoot a dart at the Warm-up – Shoot a dart at the bullseye!bullseye!

BULLSEYE

Nonlinear MotionNonlinear Motion

Projectile MotionProjectile Motion

Precision and AccuracyPrecision and Accuracy• Experimental results can be characterized by their precision and Experimental results can be characterized by their precision and

their accuracy.their accuracy.

– Precision describes the degree of exactness of a Precision describes the degree of exactness of a measurement. measurement.

• The precision of a measurement is one-half the smallest The precision of a measurement is one-half the smallest division of the instrument (ruler, graduated cylinder, etc.)division of the instrument (ruler, graduated cylinder, etc.)

• A meterstick’s smallest division is the millimeter, so you A meterstick’s smallest division is the millimeter, so you can measure the length of an object to within half a can measure the length of an object to within half a millimeter.millimeter.

– Accuracy describes how well the results of an experiment Accuracy describes how well the results of an experiment agree with the standard value.agree with the standard value.

• Instruments may need to be calibrated before a precise Instruments may need to be calibrated before a precise measurement will be accurate.measurement will be accurate.

precise but not accurate

accurate but not precise not accurate or precise

accurate and precise

correctresult

correctresult

correctresult

correctresult

Projectile MotionProjectile Motion• Projectile motion is nonlinear motion – Projectile motion is nonlinear motion –

motion along a curved path.motion along a curved path.• The object in projectile motion has two The object in projectile motion has two

independent components of motion:independent components of motion:– horizontal motionhorizontal motion– vertical motionvertical motion

Note –Note –

gravity only affects vertical gravity only affects vertical

motion.motion.


Projectile motion problems are Projectile motion problems are best solved by treating horizontal best solved by treating horizontal

and vertical motion separately.and vertical motion separately.

Vector and Scalar Quantities Vector and Scalar Quantities (Review)(Review)

• A vector quantity as both magnitude A vector quantity as both magnitude and direction.and direction.

• A scalar quantity has only magnitude.A scalar quantity has only magnitude.• Velocity is a vector, as is acceleration.Velocity is a vector, as is acceleration.• Scalars include quantities that can be Scalars include quantities that can be

specified with only magnitude such as specified with only magnitude such as mass, volume, time, etc.mass, volume, time, etc.

Velocity VectorsVelocity Vectors• An arrow is used to represent the magnitude An arrow is used to represent the magnitude

and direction of a vector quantity.and direction of a vector quantity.

• A velocity is sometimes the result of A velocity is sometimes the result of combining two or more other velocities.combining two or more other velocities.

• For example, an airplane’s velocity is a For example, an airplane’s velocity is a combination of the velocity of the airplane combination of the velocity of the airplane relative to the air and the velocity of the air relative to the air and the velocity of the air relative to the ground, or the wind velocity.relative to the ground, or the wind velocity.

• Consider the airplane show here. The Consider the airplane show here. The airplane is flying north at 100 km/h relative airplane is flying north at 100 km/h relative to the surrounding air.to the surrounding air.

– With a tailwind of 20 km/h, the plane is With a tailwind of 20 km/h, the plane is flying at a velocity of 120 km/h relative to flying at a velocity of 120 km/h relative to the ground.the ground.

– With a headwind of 20 km/h, the place is With a headwind of 20 km/h, the place is flying at a velocity of 80 km/h to the flying at a velocity of 80 km/h to the ground.ground.

Velocity Vectors (Review)Velocity Vectors (Review)

• Consider the airplane show above. The airplane is Consider the airplane show above. The airplane is flying north at 80 km/h relative to the surrounding air.flying north at 80 km/h relative to the surrounding air.

• With a crosswind of 60 km/h, the plane is flying at a With a crosswind of 60 km/h, the plane is flying at a velocity of 100 km/h relative to the ground.velocity of 100 km/h relative to the ground.

• This resultant was found using the parallelogram This resultant was found using the parallelogram method and the Pythagorean Theorem.method and the Pythagorean Theorem.

80 80 km/km/hh

60 60 km/km/hh

100 100 km/hkm/h

resultantresultant

Components of VectorsComponents of Vectors

• A velocity vector can be A velocity vector can be resolved into an equivalent resolved into an equivalent set of two component vectors set of two component vectors at right angles to each other – at right angles to each other – the horizontal component and the horizontal component and the vertical component.the vertical component.

• The resolution of a vector is The resolution of a vector is just the parallelogram method just the parallelogram method done backwards.done backwards.

Physics of Sports - SurfingPhysics of Sports - SurfingSurfing nicely illustrates component and resultant Surfing nicely illustrates component and resultant

vectors.vectors.

1.1. When surfing in the same direction as the wave, our velocity is the When surfing in the same direction as the wave, our velocity is the same as the wave’s velocity, vsame as the wave’s velocity, v. This velocity is called v. This velocity is called v because because we are moving perpendicular to the wave front.we are moving perpendicular to the wave front.

2.2. To go faster, we surf at an angle to the wave front. Now we have a To go faster, we surf at an angle to the wave front. Now we have a component of velocity parallel to the wave front, vcomponent of velocity parallel to the wave front, v llll, as well as the , as well as the perpendicular component vperpendicular component v. We can vary v. We can vary vllll, but v, but v stays relatively stays relatively constant as long as we ride the wave. Adding components, we see constant as long as we ride the wave. Adding components, we see that when surfing at an angle to the wave front our resultant that when surfing at an angle to the wave front our resultant velocity, vvelocity, vrr, exceeds v, exceeds v. .

3.3. As we increase our angle relative to the wave front, the resultant As we increase our angle relative to the wave front, the resultant velocity also increases.velocity also increases.

Projectile MotionProjectile Motion Projectiles near the surface of the earth follow a Projectiles near the surface of the earth follow a

curved path that can be resolved into horizontal curved path that can be resolved into horizontal and vertical components.and vertical components.

The horizontal component of motion for a The horizontal component of motion for a projectile is just like the horizontal motion of a projectile is just like the horizontal motion of a ball rolling freely along a level surface. ball rolling freely along a level surface. Neglecting friction, the rolling ball moves at Neglecting friction, the rolling ball moves at constant velocity.constant velocity.

The vertical component of a projectile’s velocity The vertical component of a projectile’s velocity is like the motion for a freely falling object. In the is like the motion for a freely falling object. In the vertical direction, the projectile accelerates vertical direction, the projectile accelerates downward due to gravity.downward due to gravity.

The horizontal component of motion for a The horizontal component of motion for a projectile is completely independent of the projectile is completely independent of the vertical component of motion. Their combined vertical component of motion. Their combined effects produce the variety of the curved paths of effects produce the variety of the curved paths of projectiles.projectiles.


• A dropped object and a projectile will A dropped object and a projectile will hit the ground at the same time hit the ground at the same time because gravity is the only force because gravity is the only force affecting the vertical vector.affecting the vertical vector.

Two General Types of Projectile MotionTwo General Types of Projectile Motion

1.1. Objects launched horizontallyObjects launched horizontally

2.2. Objects launched upwards at an Objects launched upwards at an angleangle

Upwardly Launched Upwardly Launched ProjectilesProjectiles

• Consider the cannonball shot at Consider the cannonball shot at an upward angle in the picture. an upward angle in the picture. Because of gravity, the cannonball Because of gravity, the cannonball follows the curved path as shown.follows the curved path as shown.

• If there were no gravity, the If there were no gravity, the cannonball would follow a cannonball would follow a straight-line path such as shown straight-line path such as shown by the dashed line.by the dashed line.

• The vertical distance the cannonball falls at any point beneath this The vertical distance the cannonball falls at any point beneath this imaginary dashed line is the same vertical distance it would fall if imaginary dashed line is the same vertical distance it would fall if it were dropped from rest and had been falling the same amount it were dropped from rest and had been falling the same amount of time.of time.

• Recall that this distance is given by d = ½gtRecall that this distance is given by d = ½gt22, where t is the , where t is the elapsed time.elapsed time.

• Rounding g to 10 m/sRounding g to 10 m/s22, at one second the cannonball is 5 m below , at one second the cannonball is 5 m below the dashed line; at 2 seconds it is 20 m below; at 3 s its 45 m the dashed line; at 2 seconds it is 20 m below; at 3 s its 45 m below, etc.below, etc.


• Note that the cannonball moves Note that the cannonball moves equal horizontal distances in equal equal horizontal distances in equal time intervals.time intervals.

• This is because there is no horizontal This is because there is no horizontal acceleration, the only acceleration is acceleration, the only acceleration is due to gravity in the vertical due to gravity in the vertical direction.direction.


• The angle and initial velocity The angle and initial velocity the projectile is launched will the projectile is launched will determine the distance the determine the distance the projectile will travel – the projectile will travel – the horizontal range.horizontal range.

• The picture shows a soccer ball The picture shows a soccer ball launched at the same initial launched at the same initial speed but at different angles.speed but at different angles.

• Notice that:Notice that:

– the soccer ball reaches different heights, the soccer ball reaches different heights,

– the paths are all parabolas, the paths are all parabolas,

– the 45° path has the longest horizontal range, and the 45° path has the longest horizontal range, and

– any two paths whose angles add up to 90° will have the same any two paths whose angles add up to 90° will have the same horizontal range (75° and 15° paths, 30° and 60° paths, etc).horizontal range (75° and 15° paths, 30° and 60° paths, etc).


• All the previous examples were All the previous examples were with negligible air resistance.with negligible air resistance.

• In the presence of air resistance, In the presence of air resistance, the path of a high speed the path of a high speed projectile falls below the idealized projectile falls below the idealized parabola and follows a solid parabola and follows a solid curve.curve.

• If air resistance is negligible, a projectile will If air resistance is negligible, a projectile will rise to its maximum height in the same time rise to its maximum height in the same time it takes to fall from that height to the ground.it takes to fall from that height to the ground.

• This is due to the constant effect of gravity. This is due to the constant effect of gravity. The deceleration due to gravity going up is The deceleration due to gravity going up is the same as the acceleration due to gravity the same as the acceleration due to gravity coming down.coming down.

• The projectile will hit the ground with the The projectile will hit the ground with the same speed it had when it was projected same speed it had when it was projected upward.upward.

v = 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

horizontalhorizontalvvxx = v cos = v cos θθ

RRxx = v = vxxtt

verticalverticalvvyy = v sin = v sin θθ

h = vh = vyyt/4t/4

t = 2vt = 2vyy/g/g

v = initial velocityv = initial velocity

θθ = launch angle = launch angle

Objects Launched at an Objects Launched at an AngleAngle

vvxx

vvyy vv

h = maximum heighth = maximum heighttt = t/2 = time in air to highest point = t/2 = time in air to highest point

RRxx = horizontal range = horizontal range

horizontalhorizontalvvxx = v cos = v cos θθ

RRxx = v = vxxtt

verticalverticalvvyy = v sin = v sin θθ

v = initial velocityv = initial velocity θθ = launch angle = launch angle

CalculationsCalculations

g = 9.8 m/sg = 9.8 m/s22

From average acceleration equations: vFrom average acceleration equations: vyy = g = gt;t; g = g = vvyy//t;t;

solve for h.solve for h.

h = ½ gh = ½ gtt22 = ½ = ½ (v(vy y / t)/ t) tt22 = ½ = ½ vvyy tt = = vvyyt/4 = h t/4 = h

h = ½ gh = ½ gtt22 = ½ v = ½ vyyt t ; t = 2v; t = 2vyy/g/g

t = total time in airt = total time in air

Important FactsImportant Facts

The 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. Only gravity effects the vertical motion.Only gravity effects the vertical motion.

Rx = horizontal range

vx = initial horizontal velocity

t = total time in the air

h = height above ground

horizontal

RRxx = v = vxxttvertical

h = ½gt2

vx

Objects Launched Objects Launched HorizontallyHorizontally

Important 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 Time is the same for both vertical and

horizontal.horizontal.

Check QuestionCheck Question

The boy on the tower throws a ball a distance of 20 m.The boy on the tower throws a ball a distance of 20 m. At what speed is the ball thrown?At what speed is the ball thrown?

• The ball is thrown horizontally, so its speed equals the The ball is thrown horizontally, so its speed equals the horizontal distance divided by the time: horizontal distance divided by the time:

v = d/t or vv = d/t or vxx = R = Rxx/t/t

• We also know that h = ½gtWe also know that h = ½gt22. Since h = 5 m, t must . Since h = 5 m, t must equal 1 s.equal 1 s.

• Solving out, we get v = d/t = 20 m / 1 s = Solving out, we get v = d/t = 20 m / 1 s = 20 m/s20 m/s

Fast-Moving Projectiles – SatellitesFast-Moving Projectiles – Satellites• For short range projectile motion such as a batted ball or a For short range projectile motion such as a batted ball or a

cannonball, we usually assume the ground is flat.cannonball, we usually assume the ground is flat.• However, for very long range projectiles the curvature of However, for very long range projectiles the curvature of

Earth’s surface must be taken into account.Earth’s surface must be taken into account.

• If an object is projected fast enough, it will If an object is projected fast enough, it will fall around the Earth and become an Earth fall around the Earth and become an Earth satellite.satellite.

• An Earth satellite, such as the space shuttle An Earth satellite, such as the space shuttle or the moon, is simply a projectile traveling or the moon, is simply a projectile traveling fast enough to fall around Earth rather than fast enough to fall around Earth rather than into it.into it.

• At the speed necessary to fall around Earth, 8 At the speed necessary to fall around Earth, 8 km/s, most objects would burn up in the km/s, most objects would burn up in the atmosphere. atmosphere.

• This is why satellites are launched at This is why satellites are launched at altitudes above 150 km – high enough not to altitudes above 150 km – high enough not to burn up but still affected by gravity.burn up but still affected by gravity.

Nonlinear MotionNonlinear Motion

Uniform Circular MotionUniform Circular Motion

Uniform Circular MotionUniform Circular Motion• Can an object be accelerated if its Can an object be accelerated if its

speed remains constant?speed remains constant?

– Yes, if the change in velocity is a change Yes, if the change in velocity is a change in the object’s direction, not its speed.in the object’s direction, not its speed.

• Uniform circular motion is the Uniform circular motion is the movement of an object or point mass movement of an object or point mass at constant speed around a circle at constant speed around a circle with a fixed axis.with a fixed axis.

– Example: a rider on a merry-go-round is Example: a rider on a merry-go-round is in uniform circular motion.in uniform circular motion.

Circular MotionCircular MotionRevolution and RotationRevolution and Rotation

• Revolution – object moves in circular Revolution – object moves in circular path around an external point.path around an external point.– Ex: Revolution around an external point: Ex: Revolution around an external point:

the Earth revolves around the sun. the Earth revolves around the sun.

• Rotation – object moves in a circular Rotation – object moves in a circular path around an internal point or axis.path around an internal point or axis.– Ex: Rotation around an axis: the Earth Ex: Rotation around an axis: the Earth

rotates (or spins) on its axis.rotates (or spins) on its axis.

RevolutionRevolutionaround an external pointaround an external point

• An object revolves at constant An object revolves at constant speed.speed.– The path is a perfect circleThe path is a perfect circle

Vectors of Circular MotionVectors of Circular Motion

The The radius for circular motion is a vector (red arrow). This radius vector locates the orbiting object. One should imagine an x, y coordinate system with its origin at the center of the circle. The radius extends from this origin to the position of the object.

The velocity vector (green arrow) shows the speed and direction of the orbiting object at all points along its path.

Note that the velocity vector is tangent to the circular path of the object and is perpendicular to the radius vector at all points on the orbit.

Centripetal ForceCentripetal Force• According to Newton’sAccording to Newton’s First Law of First Law of

MotionMotion, , an object moves in a straight an object moves in a straight line unless a force acts on it to make it line unless a force acts on it to make it turn. turn.

• An external force is necessary to An external force is necessary to make an object follow a circular path.make an object follow a circular path.

• This force is called a This force is called a centripetal centripetal ((““center seekingcenter seeking”)”) force.force.

Centripetal AccelerationCentripetal Acceleration• Since everySince every unbalanced force causes an object to unbalanced force causes an object to

accelerate in the direction of that forceaccelerate in the direction of that force ((Newton’s Second Law – F = maNewton’s Second Law – F = ma), a centripetal ), a centripetal force causes a force causes a centripetal accelerationcentripetal acceleration..

• This acceleration results from a change in This acceleration results from a change in direction, anddirection, and does not imply a change in speeddoes not imply a change in speed,, although speed may also change.although speed may also change.

• In uniform circular motion the speed does not In uniform circular motion the speed does not change and the change and the centripetal accelerationcentripetal acceleration results results only from the change in position. The centripetal only from the change in position. The centripetal acceleration vector of the object always points in acceleration vector of the object always points in toward the center of the circle (center-seeking).toward the center of the circle (center-seeking).

Examples:Examples:• Centripetal force and acceleration may be caused Centripetal force and acceleration may be caused

by:by:• friction – car rounding a curvefriction – car rounding a curve

• As a car makes a turn, the force of friction acting upon the turned wheels of the car provide the centripetal force required for circular motion.

• a rope/cord – swinging a mass on a string• As a bucket of water is tied to a string and spun

in a circle, the force of tension acting upon the bucket provides the centripetal force required for circular motion.

• gravity – planets orbiting the sun• As the Earth orbits the sun, or as the moon orbits

the Earth, the force of gravity acting upon the moon provides the centripetal force required for circular motion.

r m

vThe speed v is the

circumference divided by the period.

v = v = 22rr TT

PeriodPeriod• In all cases of uniform circular motion, a mass m In all cases of uniform circular motion, a mass m

moves in a circular path of radius r with a linear moves in a circular path of radius r with a linear (tangential) speed v. (tangential) speed v.

• The time to make one complete revolution is known The time to make one complete revolution is known as the period, T.as the period, T.

aacc == v v22//rr

and

centripetal force (N)

FFcc == mmaacc == mvv22//rr

m = mass in kg v = linear velocity in m/s

r = radius of curvature in m

centripetalacceleration

(m/s2)

Formulas:Formulas:

aacc == vv22 = = 44ππ22rr r Tr T22

and

centripetal force (N)

FFcc == mmaacc == m 44ππ22rr TT22 m = mass in kg

v = linear velocity in m/s r = radius of curvature in m

centripetalacceleration

(m/s2)

Formulas Using Period:Formulas Using Period:

The Fictitious ForceThe Fictitious Force• Centrifugal Force is a fictitious force which

is actually the absence of a centripetal force.

• It’s called fictitious because centrifugal forces exists only in rotating reference frames, not in inertial (constant velocity) reference frames.

Why Rotate a Space Station ?

Centrifugal force is a fictitious force that occurs in a rotating system. This 'force' can be used to simulate gravity in space where there is no

solid surface to enable us to feel the forces of gravity.

The centrifugal force is in a direction perpendicular to the rotation axis and radially outward. As a result the astronauts in the space station are able to walk around inside the space station as if the artificial gravity is pulling them outward away from the center of the donut shaped station.

RotationRotationaround an axisaround an axis

Rotational motion - object moves in a circular path about an internal point or axis (“rotates” or “spins”)

Angular DisplacementAngular Displacement• TheThe amount (distance) that an object

rotates is its angular displacement.• Angular displacement, θ, is given in

degrees, radians, or rotations. • 1 rotation = 360 deg = 21 rotation = 360 deg = 2ππ radians radians

θ

Degrees and RadiansDegrees and Radians• There are 360There are 360 in a circle. in a circle.

• Another common unit of Another common unit of angle measure (particularly angle measure (particularly in circles) is radians.in circles) is radians.

• There are 180There are 180 in in radians radians and 360and 360 in 2 in 2 radians. radians.

• Radians are useful when Radians are useful when dealing with calculations dealing with calculations involving revolutions. 2involving revolutions. 2 radians = 1 complete radians = 1 complete revolution.revolution.

• When working with radians, When working with radians, it is customary to work with it is customary to work with fractions of fractions of ..

Tangential SpeedTangential Speed

• Recall that linear speed is the distance moved per unit time. In circular motion, this term can be used interchangeably with the term tangential speed.

• Tangential speed is the speed of an

object moving in a circular path.

Avoid ConfusionAvoid Confusion

• Do NOT confuse Do NOT confuse tangential speedtangential speed with angular speed or rotational speedangular speed or rotational speed, , which is the number of rotations per which is the number of rotations per unit time. unit time.

• Angular speed (or velocity), Angular speed (or velocity), ωω, is given , is given in deg/s, rad/s, rpm, etc...in deg/s, rad/s, rpm, etc...

If two ladybugs sit on a rotating object at different distances from the axis, they will each have the same rotational speed but

different tangential speeds.

Example:Example:

Angular AccelerationAngular Acceleration

• An object’s angular acceleration, α, is given in deg/s2, rad/s2, rpm/s, etc...

• Formulas for rotational motion follow an exact parallel with linear motion formulas.

• The only difference is a change in variables 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 + ½ att + ½ at22

vvff22 = v = vii

22 + 2ad + 2ad

ROTATIONALROTATIONAL

ff= = ii + +tt==avavtt

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

==iitt++½ ½ tt22

ff22= = ii

22++22

Constant Acceleration Formulas

Rotational-Linear Parallels

PPEERROODDIICC

MOTIONMOTION

Periodic motion - any motion in which

the path of the object repeats itself in equal time intervals.

The simple pendulum is a great example of 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..

Other examples of periodic motion:

Bouncing ball - If you drop a ball, it will start to bounce in a regular fashion. A good rubber ball or a super-ball will keep bouncing for a long time. Because of internal friction and air resistance, the ball bounces less and less each time, until it finally stops. A perfect ball—without friction—would bounce forever.

Vibrating spring - If you start a spring vibrating, it will continue to move back-and-forth for a long time. Internal friction slows it down or dampens its vibrations.

Tuning fork - You strike a tuning fork, and you can see the ends vibrate back and forth. The vibrations cause the air to vibrate, resulting in sound or a musical note.

Circular motion - Spin a weight on a string around in circles. This is a periodic motion that repeats itself every rotation. The Earth rotates around the Sun in a periodic circular motion.

Characteristics of periodic motion

All objects that are in periodic motion have three similar characteristics: velocity, period, and amplitude.

Velocity - They all have a velocity. You can measure the velocity of a bouncing ball, the weight on a pendulum, or such.

Period - is the time the object takes to go back and forth. If you spin a weight on a string, you can measure the time it takes to go 1 revolution. Drop a ball and measure the time it takes until it bounces back up. That is its period.

Sometimes frequency is used instead of period. Frequency is the reciprocal of period. f = 1 / T

Amplitude - The amplitude is 1/2 the distance the object goes before it changes from one side of the period to the other. For an object in rotation, the amplitude is the radius of the circle (1/2 the diameter).

Learn more about projectile Learn more about projectile motionmotion

at these links:at these links:

http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l2a.htmlu3l2a.html

http://www.physicsclassroom.com/Class/vectors/U3L2a.htmlhttp://www.physicsclassroom.com/Class/vectors/U3L2a.html

http://library.thinkquest.org/2779/http://library.thinkquest.org/2779/

http://id.mind.net/~zona/mstm/physics/mechanics/curvedMotion/http://id.mind.net/~zona/mstm/physics/mechanics/curvedMotion/projectileMotion/generalSolution/generalSolution.htmlprojectileMotion/generalSolution/generalSolution.html

http://www.fortunecity.com/greenfield/eagles/180/http://www.fortunecity.com/greenfield/eagles/180/projectile_motion.htmlprojectile_motion.html

http://hyperphysics.phy-astr.gsu.edu/hbase/traconhttp://hyperphysics.phy-astr.gsu.edu/hbase/tracon

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

http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ ProjectileMotion/jarapplet.htmlProjectileMotion/jarapplet.html

http://www.msu.edu/user/brechtjo/physics/cannon/cannon.htmlhttp://www.msu.edu/user/brechtjo/physics/cannon/cannon.html

http://www.msu.edu/user/brechtjo/physics/cannon/cannon.htmlhttp://www.msu.edu/user/brechtjo/physics/cannon/cannon.html

http://library.thinkquest.org/2779/Balloon.html?tqskip1=1http://library.thinkquest.org/2779/Balloon.html?tqskip1=1

http://physics.bu.edu/~duffy/java/Projectile2.htmlhttp://physics.bu.edu/~duffy/java/Projectile2.html

http://www.physicsclassroom.com/mmedia/vectors/mzng.htmlhttp://www.physicsclassroom.com/mmedia/vectors/mzng.html

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