Transcript constant initial velocity versus how total time in air changes with angle

Four Major Types of
Two Dimensional Motion
1. Projectile Motion
2. Circular Motion
3. Rotational Motion
4. Periodic Motion
Projectile motion problems are best solved by
treating horizontal and vertical motion separately.
*IMPORTANT*
Gravity only affects vertical motion.
There are two general types
of projectile motion situations.
1. object launched horizontally
2. object launched at an angle
Object Launched Horizontally
vx = initial horizontal velocity
h(y) = initial height
above ground
t = total time in the air Rx = horizontal range
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.
horizontal
Rx = vxt
vertical
h(y)= vi + 1/2gt2
Object Launched at an Angle
v = initial velocity
q = launch angle
h(y) = maximum height
t = total time in air
Rx = horizontal range
IMPORTANT FACTS
The horizontal velocity is constant.
It rises and falls in equal time intervals.
It reaches maximum height in half the total time.
Gravity only effects the vertical motion.
horizontal
vx = v cosq
Rx = vxt
vertical
vy = v sinq
h(y)= vyt/4
t = -2vy/g
Learn more about projectile motion
at these links:
link1, link2, link3, link4, link5, link6
View projectile motion simulations at:
link1, link2, link3, link4, link5, link6
Suggested Constructivist Activities
Students use simulations to complete data tables
and make graphs of the following situations:
• constant initial velocity versus how the horizontal
range changes with angle; plot “range vs angle”
• constant initial velocity versus how total time in air
changes with angle; plot “total time vs angle”
• constant initial velocity versus how maximum height
changes with angle; plot “height vs angle”
• constant angle versus how the horizontal range
changes with initial velocity; plot “range vs velocity”
• constant angle versus how the total time in the air
changes with initial velocity; plot “time vs velocity”
• constant angle versus how the maximum height
changes with initial velocity; plot “height vs velocity”
object moves in
circular path
about an external point
(“revolves”)
According to Newton’s First Law of Motion,
objects move in a straight line unless a force
makes them turn. An external force is necessary
to make an object follow a circular path.
This force is called a
CENTRIPETAL (“center seeking”) FORCE.
Since every unbalanced force causes an object
to accelerate in the direction of that force
(Newton’s Second Law), 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.
Centripetal force and acceleration may be caused by:
•gravity - planets orbiting the sun
•friction - car rounding a curve
•a rope or cord - swinging a mass on a string
In all cases, a mass m moves in a circular path
of radius r with a linear speed v. The time to make
one complete revolution is known as the period, T.
The speed v is the
v circumference divided by the period.
r
m
v = 2pr/T
The formula for centripetal acceleration is:
ac =
2
v /r
and centripetal force is:
Fc = mac =
2
mv /r
m = mass in kg
v = linear velocity in m/s
Fc = centripetal force in N
r = radius of curvature in m
ac = centripetal acceleration in m/s2
Learn more about circular motion
at these links:
link1, link2, link3, link4, link5
View circular motion simulations at:
link1, link2, link3, link4
object moves in
circular path about
an internal point
or axis
(“rotates” or “spins”)
The amount that an object rotates is its
angular displacement.
angular displacement, q, is given in
degrees, radians, or rotations.
1 rotation = 360 deg = 2p radians
The time rate change of an object’s
angular displacement is its
angular velocity.
angular velocity, w, is given in
deg/s, rad/s, rpm, etc...
The time rate change of an object’s
angular velocity is its
angular acceleration.
Angular acceleration, a, 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
LINEAR
ROTATIONAL
vf = vi + at
d = vavt
vav = (vf + vi)/2
d = vit + 0.5at2
wf = wi + at
q = wavt
vav = (wf + wi)/2
vf2 = vi2 + 2ad
wf2 = wi2 + 2aq
q = wit +
2
0.5at
any motion in which
the path of the object
repeats itself in equal
time intervals
MOTION
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.
Learn more about pendulums and
periodic motion at these links:
link1, link2, link3, link4, link5
View pendulum simulations at:
link1, link2, link3, link4, link5