Physics 2211 Spring 2005

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Physics 2211 Lecture 07

• A Gallery of Forces
• Newtons 2nd Law of Motion
• Newtons 1st Law of Motion

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Force

• A force is a push or a pull.
• A force is caused by an agent
• and acts on an object. More
• precisely, the object and agent
• INTERACT.
• A force is a vector.

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Review Newton's Laws

• Law 1 An object subject to no external forces
  is at rest or moves with a constant
  velocity if viewed from an inertial reference
  frame.

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Weight
The Earth is the agent (on this planet)
The weight force pulls the box down
(toward the center of the Earth)
long-range force no contact needed!
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Gravity

• What is the force of gravity exerted by the earth
  on a typical physics student?

• Typical student mass m 55kg
• g 9.8 m/s2.
• Fg mg (55 kg)x(9.8 m/s2 )
• Fg 539 N (weight)

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ExampleMass vs. Weight

• An astronaut on Earth kicks a bowling ball and
  hurts his foot. A year later, the same astronaut
  kicks a bowling ball on the moon with the same
  force.
  His foot hurts...

Ouch!
(1) more (2) less (3) the same

• The masses of both the bowling ball and the
  astronaut remain the same, so his foot will feel
  the same resistance and hurt the same as before.

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ExampleMass vs. Weight
Wow! Thats light.

• However the weights of the bowling ball and the
  astronaut are less

W mgMoon gMoon lt gEarth

• Thus it would be easier for the astronaut to pick
  up the bowling ball on the Moon than on the Earth.

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The Spring Force
The spring is the agent
A compressed spring pushes on the object.
A stretched spring pulls on the object.
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Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed position
  (linear restoring force).
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality.

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Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed position
  (linear restoring force).
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality.

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Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed position
  (linear restoring force).
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality.

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Tension
The rope is the agent
The rope force (tension) is a pull along the
rope, away from the object
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Tools Ropes Strings

• Ropes strings can be used to pull from a
  distance.
• Tension (T) at a certain position in a rope is
  the magnitude of the force acting across a
  cross-section of the rope at that position.
• The force you would feel if you cut the rope and
  grabbed the ends.
• An action-reaction pair.

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Tools Ropes Strings

• Consider a horizontal segment of rope having mass
  m
• Draw a free-body diagram (ignore gravity).

• Using Newtons 2nd law (in x direction)
  FNET T2 - T1 ma
• So if m 0 (i.e., the rope is light) then T1
  ??T2

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Tools Ropes Strings

• An ideal (massless) rope has constant tension
  along the rope.

• If a rope has mass, the tension can vary along
  the rope
• For example, a heavy rope hanging from the
  ceiling...

• We will deal mostly with ideal massless ropes.

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Tools Ropes Strings

• The direction of the force provided by a rope is
  along the direction of the rope

Since ay 0 (box not moving),
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The Surface Contact Force
(one force two components)
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Surface Contact Force(microscopic view)
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There are two types of Friction
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Rolling friction
coefficient of rolling friction
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Drag
A force experienced by a body that moves through
air.
Experiment shows that to a good approximation,
Unit vector that points in the direction of the
velocity of the body
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Drag Force
DRAG occurs when an object moves in a gas or
liquid. Like friction, the force of drag always
points opposite to the direction of motion
The fluid is the agent. Skin Drag (like
friction) Form Drag (rowboat)
Drag is very small for most of the objects we
will discuss. We will always neglect drag in our
models unless we explicitly state otherwise.
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Throw a ball upward vertically
Drag decreases as the ball slows down
Drag increases as the ball speeds up slows
Drag opposes the weight as it falls
Drag adds to the weight as it rises
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Terminal velocity
At the terminal velocity, the drag force
balances the force of gravity.
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Drop a particle from rest (v 0)
As the speed (and thus drag) increases, the
slope decreases
Slope approaches zero as v approaches terminal
velocity
Without drag, v - at
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Empirical (Contact) Forces(examples)

• Linear Restoring Force
• Friction Force
• Fluid Force

Characterized by variable, experimentally determin
ed constants k, m, b, etc.
These forces are all electromagnetic in origin.
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Thrust
THRUST is a contact force exerted on rockets and
jets (and leaky balloons) by exhaust gases (the
agent).
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Electric and Magnetic Forces (Non-Contact)
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Identifying Forces---The Skier

• Identify SYSTEM

• Find contact points between SYSTEM
  ENVIRONMENTName and label the CONTACT FORCES

2. Draw PICTURE with a closed curve around the
SYSTEM. (Everything outside the curve is
the environment.

• Name and label
• LONG-RANGE FORCES

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Newtons 2nd Law of Motion Newtons 1st Law of
Motion
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Newtons First Law
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Problem Accelerometer

• A weight of mass m is hung from the ceiling of a
  car with a massless string. The car travels on a
  horizontal road, and has an acceleration a in the
  x direction. The string makes an angle ? with
  respect to the vertical (y) axis. Solve for ? in
  terms of a and g.

?

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Accelerometer

• Draw a free body diagram (FBD) for the mass
• What are all of the forces acting?

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Accelerometer

• Resolve forces into components

• Sum forces in each dimension separately

• Eliminate T

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Accelerometer

• Say the car goes from 0 to 60 mph uniformly in 10
  seconds
• 60 mph (60 x 0.45) m/s 27 m/s.
• Acceleration a ?v/?t 2.7 m/s2.
• So a/g 2.7 / 9.8 0.28 .
• ? arctan (a/g) 15.6 deg

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Problem Inclined plane

• A block of mass m slides down a frictionless ramp
  that makes angle ? with respect to the
  horizontal. What is its acceleration a ?

m

?
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Inclined plane

• Define convenient axes parallel and perpendicular
  to plane
• Acceleration a is in x direction only.

m

?
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Inclined plane

• Draw a FBD.

• Resolve forces into components sum forces in x
  and y directions separately

Assume forces are acting at center of mass of
block.
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Angles of an Inclined plane
Lines are perpendicular, so the angles are the
same!
?
?
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Vocabulary

• A Reference Frame is the (x,y,z) coordinate
  system you choose for making measurements.
• An Inertial Reference Frame is a frame where
  Newtons Laws
• are valid.
• USE NEWTONS LAWS TO TEST
• FOR INERTIAL REFERENCE FRAMES

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Test for Inertial Reference Frame
EXAMPLE Airplane parked on runway A ball is
placed on the floor of the plane no net forces
act on the ball.
Less obvious a plane cruising at constant
velocity is also an inertial
reference frame!

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Test for Inertial Reference Frame

• EXAMPLE Airplane taking off.
• Ball placed on floor of plane no net forces act
  on ball.
• .

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We live in a World of Approximations . . .
Strictly speaking, an Inertial Reference Frame
has zero acceleration with respect to the
distant stars.
The Earth accelerates a little (compared to the
distant stars) due to its daily rotation and its
yearly revolution around the Sun. Nevertheless,
to a good approximation, the Earth is an
Inertial Reference Frame.
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Is Atlanta a good IRF?

• Is Atlanta accelerating?
• YES!

• Atlanta is on the Earth.
• The Earth is rotating.

• What is the acceleration (centripetal) of Atlanta?

• T 1 day 8.64 x 104 sec,
• R RE 6.4 x 106 meters .
• Plug this in aatl 0.034 m/s2 ( 1/300 g)
• Close enough to zero that we will ignore it.
• Atlanta is a pretty good IRF.

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Physics 2211 Lecture 12

• 2-D, 3-D Kinematics and Projectile Motion
• Independence of x and y components
• Georgia Tech track and field example
• Football example
• Shoot the monkey

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3-D Kinematics

• The position, velocity, and acceleration of a
  particle in 3 dimensions can be expressed as

• We have already seen the 1-D kinematics equations

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3-D Kinematics

• For 3-D, we simply apply the 1-D equations to
  each of the component equations.

• Which can be combined into the vector equations

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3-D Kinematics

• So for constant acceleration we can integrate to
  get

• Aside the 4th kinematics equation can be
  written as

• (more on this later)

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2-D Kinematics

• Most 3-D problems can be reduced to 2-D problems
  when acceleration is constant
• Choose y axis to be along direction of
  acceleration
• Choose x axis to be along the other direction
  of motion
• Example Throwing a baseball (neglecting air
  resistance)
• Acceleration is constant (gravity)
• Choose y axis up ay -g
• Choose x axis along the ground in the direction
  of the throw

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Uniform Circular Motion

• What does it mean?
• How do we describe it?
• What can we learn about it?

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What is Uniform Circular Motion?

• Motion with
• Constant Radius R
• (Circular)
• Constant Speed
• (Uniform)

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How can we describe Uniform Circular Motion?

• In general, one coordinate system is as good as
  any other
• Cartesian
• (x,y) position
• (vx ,vy) velocity
• Polar
• (R,? ) position
• (vR ,? ) velocity
• In uniform circular motion
• R is constant (hence vR 0).
• ? (angular velocity) is constant.
• Polar coordinates are a natural way to describe
  Uniform Circular Motion!

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Polar Coordinates
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Angular Motion

• The arc length s (distance along the
  circumference) is related to the angle in a
  simple way
• s R?, where ? is the angular displacement.
• units of ? are called radians.
• For one complete revolution
• 2?R R?c
• ?c 2?
• ?? has a period 2?.
• 1 revolution 2??radians

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Angular Motion

• In Cartesian coordinates, we say velocity vx
  dx/dt.
• x vxt (vx constant)
• In polar coordinates, angular velocity d?/dt ?.
• ? ? t (? constant)
• ? has units of radians/second.
• Displacement s v t.
• but s R? R? t, so

v R ?
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Aside Period and Frequency

• Recall that 1 revolution 2? radians
• (a) frequency ( f ) revolutions / second
• (b) angular velocity ( ? ) radians / second
• By combining (a) and (b)
• ? (rad/s) 2? rad/rev x f (rev/s)
• ? 2?f
• Realize that
• period (T) seconds / revolution
• So T 1 / f 2?/?

? 2? / T 2?f
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Summary

• Relationship between Cartesian and Polar
  coordinates

• Angular motion

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Polar Unit Vectors

• We are familiar with the Cartesian unit vectors
• Now introduce polar unit-vectors and
  
• points in radial direction
• points in tangential direction
  (counterclockwise)

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Acceleration in Uniform Circular Motion

• Even though the speed is constant, velocity is
  not constant since the direction is changing
  acceleration is not zero!

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Acceleration in Uniform Circular Motion

• This is called Centripetal Acceleration.
• Now lets calculate the magnitude

But ?R v ?t for small ?t
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Centripetal Acceleration

• Uniform Circular Motion results in acceleration
• Magnitude a v2 / R
• Direction - r (toward center of circle)

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Centripetal Acceration(in terms of w )
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ExampleUniform Circular Motion

• A fighter pilot flying in a circular turn will
  pass out if the centripetal acceleration he
  experiences is more than about 9 times the
  acceleration of gravity g. If his F18 is moving
  with a speed of 300 m/s, what is the approximate
  diameter of the tightest turn this pilot can make
  and survive to tell about it ?
• (1) 500 m
• (2) 1000 m
• (3) 2000 m

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ExampleSolution
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Example Propeller Tip

• The propeller on a stunt plane spins with
  frequency f 3500 rpm. The length of each
  propeller blade is L 80 cm. What centripetal
  acceleration does a point at the tip of a
  propeller blade feel?

f
what is a here?
L
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Example

• First calculate the angular velocity of the
  propeller

• so 3500 rpm means ? 367 s-1
• Now calculate the acceleration.
• a ?2R (367s-1)2 x (0.8m) 1.1 x 105 m/s2
  11,000 g
• direction of acceleration points towards the
  propeller hub.

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Example Newton the Moon

• What is the acceleration of the Moon due to its
  motion around the Earth?
• What we know (Newton knew this also)
• T 27.3 days 2.36 x 106 s (period 1 month)
• R 3.84 x 108 m (distance to moon)
• RE 6.35 x 106 m (radius of earth)

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Moon

• Calculate angular velocity

• So ? 2.66 x 10-6 s-1.
• Now calculate the acceleration.
• a ?2R 0.00272 m/s2 0.000278 g
• direction of acceleration points is towards the
  center of the Earth.

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Moon

• So we find that amoon / g 0.000278
• Newton noticed that RE2 / R2 0.000273
• This inspired him to propose that FMm ? 1 / R2

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ExampleCentripetal Acceleration

• The Space Shuttle is in Low Earth Orbit (LEO)
  about 300 km above the surface. The period of
  the orbit is about 91 min. What is the
  acceleration of an astronaut in the Shuttle in
  the reference frame of the Earth?
  (The radius of the
  Earth is 6.4 x 106 m.)
• (1) 0 m/s2
• (2) 8.9 m/s2
• (3) 9.8 m/s2

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ExampleCentripetal Acceleration

• First calculate the angular frequency ?

• Realize that

RO RE 300 km 6.4 x 106 m 0.3 x 106 m
6.7 x 106 m
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ExampleCentripetal Acceleration

• Now calculate the acceleration

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Physics 2211 Lecture 16

• Circular Orbits
• Fictitious forces

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SEEM to be very different, BUT they have the same
free body diagram . . .
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Orbital Motion is Projectile Motion!
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Orbiting projectile is in free fall
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Fictitious forces in Non-Inertial Frames of
Reference
A car and driver move a constant speed. Suddenly,
the driver brakes
Outside observer if the seat is
frictionless, the driver continues forward at
constant speed and collides with the front
window.
Inside observer a force F throws the driver
against the front window.
F
F is fictitious the observer is in a
non- inertial (accelerated) frame where Newtons
laws do not apply. There is no true push or
pull from anything.
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Fictitious forces in Non-Inertial Frames of
Reference

• Car turns
• Book continues on straight line
• In drivers reference frame,
• apparent (fictitious) force moves book
• across the bench seat

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A car is passing over the top of a hill at
(non-zero) speed v. At this instant,

• 1 n gt w
• 2 n w
• 3 n lt w
• 4 Cant tell without knowing v

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3
A ball on a string swings in a vertical circle.
The string breaks when the string is horizontal
and the ball is moving straight up. Which
trajectory does the ball follow thereafter?
2
4
1
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Roller-Coaster
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Full Disclosure
mg
N
N
N
N
a
mg
mg
a
mg
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Example
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Example

• Treat horizontal motion and vertical motion
  separately

• Then just add the results Principle of
  Superposition

• Velocity at highest point

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Example

• Vertical Motion

• Need to determine magnitude of take-off velocity

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Example

• Magnitude of take-off velocity

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Example

• Two footballs are thrown from the same point on a
  flat field. Both are thrown at an angle of 30o
  above the horizontal. Ball 2 has twice the
  initial speed of ball 1. If ball 1 is caught a
  distance D1 from the thrower, how far away from
  the thrower D2 will the receiver of ball 2 be
  when he catches it?

• (1) D2 2D1 (2) D2 4D1 (3) D2 8D1

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Example

• The horizontal distance a ball will go is simply
  x (horizontal speed) x (time in air) v0x t

• To figure out time in air, consider the
  equation for the height of the ball

• When the ball is caught, y y0

(time of catch)
(time of throw)
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Example

• So the time spent in the air is

• The range, R, is thus

• Ball 2 will go 4 times as far as ball 1!

• Notice For maximum range,

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Shooting the Monkey(tranquilizer gun)

• Where does the zookeeper aim if he wants to hit
  the monkey?
• ( He knows the monkey willlet go as soon as he
  shoots ! )

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Shooting the Monkey

• If there were no gravity, simply aim

at the monkey
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Shooting the Monkey

• With gravity, still aim at the monkey!

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Shooting the Monkey
x v0 t y -1/2 g t2

• This may be easier to think about.
• Its exactly the same idea!!

x x0 y -1/2 g t2