Class 4.1

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Class 4.1

• Analysis of Motion
• Newtons Laws

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Class Objective

• Learn and apply Newtons First, Second, and Third
  Laws
• Unidirectional
• Multidirectional
• Learn the relationship between position,
  velocity, and acceleration

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RAT 1
4
Some Definitions (1D)
Position - location on a straight line
x
Displacement - change in location on a
straight line
Dx x2 - x1 4 - (-2) 6
x
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Some Definitions (1D)
Average Velocity - rate of position change with
time (vector)
Instantaneous Velocity (vector)
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Average and Instantaneous Velocity (1D)
Position
x2
x2 - x1 t2 - t1
Slope Average Velocity
x1
t1 t2
Time
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Some Definitions (2D)

• Position -- a location usually described by a
  graphic on a map or by a coordinate system

4
3
2
(4, 3)
1
0
0
5
4
3
2
1
-3
-2
-1
-1
-2
-3
(-2, -3)
-4
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Some Definitions (2D)

• Displacement -- change in position,
• where

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(4, 3)
3
2
1
0
0
1
2
3
4
-1
-2
-3
5
-1
-2
-3
-4
(-2, -3)
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Some Definitions (2D)

• Average velocity (vector)
• Instantaneous velocity (vector)
• Speed - the magnitude of instantaneous velocity
  (scalar)

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Some Definitions (1D)
Average Acceleration - rate of velocity change
with time (vector)
Instantaneous Acceleration (vector)
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Average and Instantaneous Acceleration (1D)
Velocity
v2
v2 - v1 t2 - t1
Slope Average
Acceleration
v1
t1 t2
Time
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Some Definitions (2D)

• Average Acceleration (vector)
• Instantaneous Acceleration (vector)

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Paired Exercise 1

• What is the distance traveled?
• What is the acceleration at 1.25 hours?

Speed, miles per hour
Time, hours
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For constant acceleration...

• if acceleration is constant
• integrating both sides
• v0 is the original value at the beginning of the
  time interval

(Definition)
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Constant Acceleration

• substituting the velocity equation from the
  previous page
• integrating both sides
• yields

(Definition)
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Equations of Motion (Constant Acceleration)

• Velocity
• Position (in terms of x)

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Multiple Directions

• Equations of motion can be written for each
  direction independently.
• Velocity
• Position

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Distance, Velocity, and Acceleration

• Suppose a dragster has constant acceleration.
• If a dragster starts from rest and accelerates to
  60 mph in 10 seconds. How far did it travel?

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Plot Speed vs time
What does the area under the line represent?
60 mph
(1 mi/min)
speed
time
10 seconds
(1/6 min)
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Distances.
Area distance? Sure
Right?
So
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Your Turn
RAT 2
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Momentum
v
m
momentum
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Newtons 1st Law

• Every body persists in its state of rest or of
  uniform motion in a straight line unless it is
  compelled to change that state by forces
  impressed upon it.
• In other words
• In the absence of a net force applied to an
• object, momentum stays constant.

Newton
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Newtons Second Law

• The time-rate-of-change of momentum is
  proportional to the net force on the object.
• If mass is constant...

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Gravity Force
Fmamg
m

32.1740 ft/s2
Standard g
9.80665 m/s2
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Newtons Second Law

• If mass is NOT constant then

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Newtons Third Law

• To every action there is always opposed an equal
  reaction or, the mutual actions of two bodies
  upon each other are always equal and directed in
  contrary parts.
• 
  Newton

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Newtons Third Law

• Other statements
• Forces always exist by the interaction of two (or
  more) bodies
• The force on one body is equal and opposite to
  the force on another body
• It is impossible to have a single isolated force
  acting in one direction
• The designation of an action force and a
  reaction force is arbitrary because there is
  mutual interaction between the bodies

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Newtons Third Law

• A Consequence
• The earth and the moon orbit about a common
  point about 1000 miles below the surface of the
  earth because the earth pulls on the moon and the
  moon pulls on the earth.

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Example Newtons 3rd Law

• Consider a rocket with constant exhaust gas
  velocity
• The mass changes (obviously) as the fuel is
  burned and the gas is ejected.

Positive
fuel
ve
v
m
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Example Newtons 3rd Law

• The magnitude of the net force acting on the
  rocket can be determined by observing its
  acceleration
• where m is the instantaneous mass of the rocket
  and dv is the instantaneous change in rocket
  velocity.

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Example Newtons 3rd Law

• The magnitude of the net force acting on the
  ejected gas is
• where ve is the velocity of ejected gas and
  dm/dt is the rate mass is ejected from the rocket.

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Example Newtons 3rd Law

• From Newtons 3rd Law, these two forces must be
  opposite and equal to each other, so
• or,

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Example 3rd Law

• Using calculus, this can be solved to yield
• where m0 is the initial mass of rocket including
  fuel

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Pair Exercise 21 - Newtons Laws

• A pickup truck is moving with a constant speed of
  30 mph.
• You are sitting in the back of the truck and you
  throw a ball straight upwards at a speed of 20
  mph.
• More

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Pair Exercise 2 (cont)

• Neglecting air resistance
• 1. How long does it take the ball to reach
  maximum height?
• 2. What is the maximum height of the ball?
• 3. Using Excel, calculate the vertical height and
  horizontal distance as a function of time. Plot
  the vertical height as a function of horizontal
  distance.
• 4. Where will the ball land with respect to the
  pickup truck?

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Pair Exercise 3

• As a PAIR, take 15 minutes to solve the following
  problem
• A cannon fires a projectile with an initial
  velocity of 1500 ft/s at a 50o angle. Assume
  that the projectile lands at a point that is 120
  ft below that at which it was launched. NEGLECT
  DRAG DUE TO AIR RESISTANCE.

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Pair Exercise 3 (cont)

• Using Excel determine
• Time of flight
• Range (x coord. at point of impact)
• Maximum height of projectile
• Using Excel, plot trajectory (y vs. x) of
  projectile. Use at least 100 data points from
  point of launch to point of impact.
• SAVE YOUR FILE IT MIGHT BE USEFUL.

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Why Newtons Laws?

• Engineers use models to predict things such as
  motion, fluid flow, lift on an airplane wing,
  movement of neutrons in a nuclear reactor,
  deflection of beams or columns, etc.
• Newtons laws are widely used and a good first
  example of engineering models.

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More on Models

• Question If I toss a piece of chalk at a
  sleeping student, does its path follow a
  parabola?
• Answer Not exactly, because air resistance
  affects the motion. Also, we should consider the
  effect of the spinning earth as it moves around
  the sun in an ellipse. However, for most
  practical work, a parabola is close enough.

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Homework 04.1

• Due in one week