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Warm-up – Shoot a dart at the bullseye!
BULLSEYE
Nonlinear Motion
Projectile Motion
Precision and Accuracy
• Experimental results can be characterized by their precision and their
accuracy.
– Precision describes the degree of exactness of a measurement.
• The precision of a measurement is one-half the smallest
division of the instrument (ruler, graduated cylinder, etc.)
• A meterstick’s smallest division is the millimeter, so you can
measure the length of an object to within half a millimeter.
– Accuracy describes how well the results of an experiment agree
with the standard value.
• Instruments may need to be calibrated before a precise
measurement will be accurate.
correct
result
correct
result
correct
result
correct
result
precise but not accurate
accurate but not precise
not accurate or precise
accurate and precise
Projectile Motion
• Projectile motion is nonlinear motion –
motion along a curved path.
• The object in projectile motion has two
independent components of motion:
– horizontal motion
– vertical motion
Projectile Motion
Projectile motion problems are best
solved by treating horizontal and
vertical motion separately.
Note –
gravity only affects vertical motion.
Vector and Scalar Quantities
(Review)
• A vector quantity as both magnitude
and direction.
• A scalar quantity has only magnitude.
• Velocity is a vector, as is acceleration.
• Scalars include quantities that can be
specified with only magnitude such as
mass, volume, time, etc.
Velocity Vectors
• An arrow is used to represent the magnitude
and direction of a vector quantity.
• A velocity is sometimes the result of combining
two or more other velocities.
• For example, an airplane’s velocity is a
combination of the velocity of the airplane
relative to the air and the velocity of the air
relative to the ground, or the wind velocity.
• Consider the airplane show here. The airplane
is flying north at 100 km/h relative to the
surrounding air.
– With a tailwind of 20 km/h, the plane is
flying at a velocity of 120 km/h relative to
the ground.
– With a headwind of 20 km/h, the place is
flying at a velocity of 80 km/h to the
ground.
Velocity Vectors (Review)
80 km/h
100 km/h
resultant
60 km/h
• Consider the airplane show above. The airplane is
flying north at 80 km/h relative to the surrounding air.
• With a crosswind of 60 km/h, the plane is flying at a
velocity of 100 km/h relative to the ground.
• This resultant was found using the parallelogram
method and the Pythagorean Theorem.
Components of Vectors
• A velocity vector can be
resolved into an equivalent set
of two component vectors at
right angles to each other –
the horizontal component and
the vertical component.
• The resolution of a vector is
just the parallelogram method
done backwards.
Physics of Sports - Surfing
Surfing nicely illustrates component and resultant vectors.
1.
When surfing in the same direction as the wave, our velocity is the same
as the wave’s velocity, v. This velocity is called v because we are
moving perpendicular to the wave front.
2.
To go faster, we surf at an angle to the wave front. Now we have a
component of velocity parallel to the wave front, vll, as well as the
perpendicular component v. We can vary vll, but v stays relatively
constant as long as we ride the wave. Adding components, we see that
when surfing at an angle to the wave front our resultant velocity, vr,
exceeds v.
3.
As we increase our angle relative to the wave front, the resultant
velocity also increases.
Projectile Motion
 Projectiles near the surface of the earth follow a
curved path that can be resolved into horizontal and
vertical components.
 The horizontal component of motion for a projectile
is just like the horizontal motion of a ball rolling
freely along a level surface. Neglecting friction, the
rolling ball moves at constant velocity.
 The vertical component of a projectile’s velocity is
like the motion for a freely falling object. In the
vertical direction, the projectile accelerates
downward due to gravity.
 The horizontal component of motion for a projectile
is completely independent of the vertical component
of motion. Their combined effects produce the
variety of the curved paths of projectiles.
Projectile Motion
• A dropped object and a projectile will
hit the ground at the same time because
gravity is the only force affecting the
vertical vector.
Two General Types of Projectile Motion
1. Objects launched horizontally
2. Objects launched upwards at an angle
Upwardly Launched Projectiles
• Consider the cannonball shot at an
upward angle in the picture.
Because of gravity, the cannonball
follows the curved path as shown.
• If there were no gravity, the
cannonball would follow a straightline path such as shown by the
dashed line.
• The vertical distance the cannonball falls at any point beneath this
imaginary dashed line is the same vertical distance it would fall if it
were dropped from rest and had been falling the same amount of
time.
• Recall that this distance is given by d = ½gt2, where t is the elapsed
time.
• Rounding g to 10 m/s2, 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 below, etc.
Upwardly Launched Projectiles
• Note that the cannonball moves equal
horizontal distances in equal time
intervals.
• This is because there is no horizontal
acceleration, the only acceleration is
due to gravity in the vertical direction.
Upwardly Launched Projectiles
• The angle and initial velocity the
projectile is launched will
determine the distance the
projectile will travel – the
horizontal range.
• The picture shows a soccer ball
launched at the same initial speed
but at different angles.
• Notice that:
– the soccer ball reaches different heights,
– the paths are all parabolas,
– the 45° path has the longest horizontal range, and
– any two paths whose angles add up to 90° will have the same horizontal
range (75° and 15° paths, 30° and 60° paths, etc).
Upwardly Launched Projectiles
• All the previous examples were with
negligible air resistance.
• In the presence of air resistance,
the path of a high speed projectile
falls below the idealized parabola
and follows a solid curve.
• If air resistance is negligible, a projectile will
rise to its maximum height in the same time it
takes to fall from that height to the ground.
• This is due to the constant effect of gravity.
The deceleration due to gravity going up is the
same as the acceleration due to gravity coming
down.
• The projectile will hit the ground with the same
speed it had when it was projected upward.
Objects Launched at an Angle
vy
v
h = maximum height
vx
v = initial velocity
θ = launch angle
v = initial velocity
horizontal
vx = v cos θ
Rx = vxt
Rx = horizontal range
t = total time in air
qvertical
= launch angle
vy = v sin θ
h = vyt/4
t = 2vy/g
Calculations
h = maximum height t = t/2 = time in air to highest point
v = initial velocity θ = launch angle t = total time in air
Rx = horizontal range
horizontal
vx = v cos θ
Rx = vxt
g = 9.8 m/s2
vertical
vy = v sin θ
From average acceleration equations: vy = gt; g = vy/t;
solve for h.
h = ½ gt2 = ½ (vy / t) t2 = ½ vy t = vyt/4 =
h = ½ gt2 = ½ vyt ; t = 2vy/g
h
Important Facts
 The horizontal velocity is constant.
 It rises and falls in equal time intervals.
 It reaches maximum height in half the total time.
 Only gravity effects the vertical motion.
Objects Launched Horizontally
Rx = horizontal range
vx
vx = initial horizontal velocity
t = total time in the air
h = height
above ground
horizontal
Rx = vxt
vertical
h = ½gt2
Important Facts
 There is no horizontal acceleration.
 There is no initial vertical velocity.
 The horizontal velocity is constant.
 Time is the same for both vertical and
horizontal.
Check Question
The boy on the tower throws a ball a distance of 20 m.
At what speed is the ball thrown?
• The ball is thrown horizontally, so its speed equals the
horizontal distance divided by the time:
v = d/t or vx = Rx/t
• We also know that h = ½gt2. Since h = 5 m, t must equal 1 s.
• Solving out, we get v = d/t = 20 m / 1 s = 20 m/s
Fast-Moving Projectiles – Satellites
• For short range projectile motion such as a batted ball or a
cannonball, we usually assume the ground is flat.
• However, for very long range projectiles the curvature of Earth’s
surface must be taken into account.
• If an object is projected fast enough, it will fall
around the Earth and become an Earth satellite.
• An Earth satellite, such as the space shuttle or
the moon, is simply a projectile traveling fast
enough to fall around Earth rather than into it.
• At the speed necessary to fall around Earth, 8
km/s, most objects would burn up in the
atmosphere.
• This is why satellites are launched at altitudes
above 150 km – high enough not to burn up but
still affected by gravity.
Nonlinear Motion
Uniform Circular Motion
Uniform Circular Motion
• Can an object be accelerated if its
speed remains constant?
– Yes, if the change in velocity is a change in
the object’s direction, not its speed.
• Uniform circular motion is the
movement of an object or point mass at
constant speed around a circle with a
fixed axis.
– Example: a rider on a merry-go-round is in
uniform circular motion.
Circular Motion
Revolution and Rotation
• Revolution – object moves in circular
path around an external point.
– Ex: Revolution around an external point:
the Earth revolves around the sun.
• Rotation – object moves in a circular
path around an internal point or axis.
– Ex: Rotation around an axis: the Earth
rotates (or spins) on its axis.
Revolution
around an external point
• An object revolves at constant speed.
– The path is a perfect circle
Vectors of Circular Motion
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 Force
• According to Newton’s First Law of
Motion, an object moves in a straight line
unless a force acts on it to make it turn.
• An external force is necessary to make
an object follow a circular path.
• This force is called a centripetal (“center
seeking”) force.
Centripetal Acceleration
• Since every unbalanced force causes an object to
accelerate in the direction of that force (Newton’s
Second Law – F = ma), a centripetal force causes a
centripetal acceleration.
• This acceleration results from a change in direction,
and does not imply a change in speed, although speed
may also change.
• In uniform circular motion the speed does not change
and the centripetal acceleration results only from
the change in position. The centripetal acceleration
vector of the object always points in toward the
center of the circle (center-seeking).
Examples:
• Centripetal force and acceleration may be caused by:
• friction – 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.
Period
• In all cases of uniform circular motion, a mass m moves
in a circular path of radius r with a linear (tangential)
speed v.
• The time to make one complete revolution is known as
the period, T.
v
r
m
The speed v is the
circumference divided
by the period.
v = 2pr
T
Formulas:
centripetal
acceleration
(m/s2)
and
centripetal force
(N)
ac =
2
v /r
Fc = mac
2
= mv /r
m = mass in kg
v = linear velocity in m/s
r = radius of curvature in m
Formulas Using Period:
centripetal
acceleration
(m/s2)
and
centripetal force
(N)
ac = v2 = 4π2r
r
T2
Fc = mac = m 4π2r
T2
m = mass in kg
v = linear velocity in m/s
r = radius of curvature in m
The 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.
Rotation
around an axis
Rotational motion - object moves in a
circular path about an internal point
or axis (“rotates” or “spins”)
Angular Displacement
• The amount (distance) that an object
rotates is its angular displacement.
• Angular displacement, θ, is given in
degrees, radians, or rotations.
• 1 rotation = 360 deg = 2π radians
θ
Degrees and Radians
• There are 360 in a circle.
• Another common unit of angle
measure (particularly in
circles) is radians.
• There are 180 in p radians
and 360 in 2p radians.
• Radians are useful when
dealing with calculations
involving revolutions. 2p
radians = 1 complete
revolution.
• When working with radians, it
is customary to work with
fractions of p.
Tangential 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 Confusion
• Do NOT confuse tangential speed with
angular speed or rotational speed, which
is the number of rotations per unit time.
• Angular speed (or velocity), ω, is given in
deg/s, rad/s, rpm, etc...
Example:
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.
Angular 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.
Constant
Constant Acceleration Formulas
LINEAR
ROTATIONAL
vf = vi + at
d = vavt
vav = (vf + vi)/2
2
d = vit + ½ at
2
vf =
2
vi
+ 2ad
wf = wi + at
q = wavt
vav = (wf + wi)/2
q = wit + ½
2
at
wf2 = wi2 + 2aq
Rotational-Linear Parallels
MOTION
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 period, T, of a simple pendulum
(time needed for one complete cycle)
is approximated by the equation:
l
T  2p
g
where l is the length of the pendulum
and g 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 motion
at these links:
http://www.glenbrook.k12.il.us/gbssci/phys/Class/vectors/u3l2a.html
http://www.physicsclassroom.com/Class/vectors/U3L2a.html
http://library.thinkquest.org/2779/
http://id.mind.net/~zona/mstm/physics/mechanics/curvedMotion/
projectileMotion/generalSolution/generalSolution.html
http://www.fortunecity.com/greenfield/eagles/180/projectile_motion.
html
http://hyperphysics.phy-astr.gsu.edu/hbase/tracon
View projectile motion simulations at:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/
ProjectileMotion/jarapplet.html
http://www.msu.edu/user/brechtjo/physics/cannon/cannon.html
http://www.msu.edu/user/brechtjo/physics/cannon/cannon.html
http://library.thinkquest.org/2779/Balloon.html?tqskip1=1
http://physics.bu.edu/~duffy/java/Projectile2.html
http://www.physicsclassroom.com/mmedia/vectors/mzng.html