Chapter 2: Motion Part 1

1
Chapter 2 Motion Part 1

• Alyssa Jean-Mary

2
Speed

• The speed of something is the rate at which it
  covers distance
• If the speed is high, it travels faster and it
  covers more distance in a given period of time
• Speed (v) distance (d) / time (t)
• Common units to express speed are meters/second
  (m/s)
• When we dont know how something moved, its speed
  is referred to as the average speed i.e. it
  might have moved faster at some times and slower
  at other times
• For example, if a car has an average speed of 50
  mi/h, it doesnt mean that it was moving at a
  constant speed of 50 mi/h during the entire time
  when it slowed down for traffic or stopped at a
  stop sign, it was obviously moving slower than 50
  mi/h
• The instantaneous speed is how fast something is
  going at any given moment
• In a car, what the speedometer reads is the
  instantaneous speed of the car

3
Steps to Solving a Problem

• Step 1 Identify the information that is given in
  the problem.
• Step 2 Identify what information the problem is
  looking for.
• Step 3 Identify the equation that is needed to
  obtain the information you are looking for, while
  using the information you were given.
• Step 4 Put the given information into the
  equation found in Step 3 and solve for the
  information you are looking for.

4
Example Calculation of Speed

• Example What is the speed of an object that
  traveled 34 meters in 65 seconds?
• Answer
• 1. Given 34 meters, 65 seconds
• 2. Looking for speed
• 3. Equation v d/t
• 4. Solution v d/t 34 meters/65 seconds
  0.52 m/s

5
Distance and Time

• The equation for speed can be rewritten in order
  to calculate
• the distance (d) something travels at a given
  speed (v) in a given period of time (t)
• distance (d) speed (v) x time (t)
• the time (t) something takes to travel a given
  distance (d) going a given speed (v)
• time (t) distance (d) / speed (v)
• Distance is commonly expressed in meters (m) and
  time is commonly expressed in seconds (s)

6
Distance Solving the Equation and Example
Calculation

• Solving the Equation
• 1. Start with the equation for speed v d/t
• 2. Get d on a side by itself by multiplying both
  sides by t, thus obtaining the equation for
  distance d vt

• Example How far did an object move in 76 seconds
  if it was traveling at a speed of 2.5 m/s?
• Answer
• 1. Given 76 seconds, 2.5m/s
• 2. Looking for how far distance
• 3. Equation d vt
• 4. Solution d vt 2.5 m/s x 76 s 190 m

7
Time Solving the Equation and Example Calculation

• Solving the Equation
• 1. Start with the equation for speed v d/t
• 2. Get t on top (it is now on the bottom) by
  multiplying both sides by t, thus the equation
  is vt d
• 3. Get t on a side by itself by dividing both
  sides by v, thus obtaining the equation for time
  t d/v

• Example How long did it take for an object move
  230 meters if it was traveling at a speed of 14.7
  m/s?
• Answer
• 1. Given 230 meters, 14.7m/s
• 2. Looking for how long time
• 3. Equation t d/v
• 4. Solution t d/v 230 m / 14.7 m/s 15.6 s

8
Scalar Quantities and Vector Quantities

• Scalar quantities are quantities that contain a
  number and a unit. Scalar quantities give only
  the magnitude (how large it is), not the
  direction
• Examples of scalar quantities
• Speed
• Vector quantities, on the other hand, are
  quantities that contain both a magnitude and a
  direction
• Examples of vector quantities
• Displacement, which is change in position
• Force
• Velocity, which is the speed of something and the
  direction of this speed

9
Representing Vector Quantities

• Vector quantities are represented by a vector, a
  straight line that has an arrowhead at one end to
  show the direction of the quantity
• The length of the line is scaled for the
  magnitude of the quantity
• Vector quantities are usually printed in bold
  (i.e. Force F), while scalar quantities are
  usually printed in italics (i.e. Speed v). To
  represent the magnitude of a vector quantity,
  italics is also used (i.e. the magnitude of a
  Force, F, is F). When a vector quantity is
  handwritten, an arrow over the symbol is used to
  indicate that it is a vector quantity.

10
Adding Scalar Quantities and Vector Quantities

• To add scalar quantities, ordinary arithmetic is
  used
• To add vector quantities,
• if they are going in the same direction, ordinary
  arithmetic is used
• if they are not going in the same direction, the
  following steps need to be followed
• 1. Draw the vectors that are to be added together
  tail to head.
• In this example, to add vector B to vector A,
  draw vector B with its tail at the head of vector
  A
• 2. Draw a vector from the tail of the first
  vector to the head of the last vector to be added
  this vector is the addition of all the vectors
• 2. Draw a vector C from the tail of vector A to
  the head of vector B vector C is the addition
  of vector B to vector A

11
Pythagorean Theorem

• A right triangle is a triangle in which two of
  its sides are perpendicular to each other in
  other words, its when the two sides meet at a
  90 angle
• The Pythagorean Theorem states that, in a right
  triangle, the square of each of the short sides
  added together is equal to the square of the
  longest side (i.e. the hypotenuse)
• If the short sides are A and B and the long side
  is C, the Pythagorean Theorem can be shown as the
  following equation
• A2 B2 C2
• When the Pythagorean Theorem is applied to adding
  vectors, A, B, and C are the magnitudes of the
  vectors A, B, and C
• The following equations can be used to find one
  of the sides if the length of the other two sides
  are known
• A v(C2 B2)
• B v(C2 A2)
• C v(A2 B2)

12
Example Calculations Using the Pythagorean Theorem

• Example 1 What is the length of side A if side B
  is 2.3 m and side C is 7.6 m?
• Answer
• 1. Given side B 2.3m, side C 7.6m
• 2. Looking for length of side A
• 3. Equation A v(C2 B2)
• 4. Solution A v(C2 B2) v(7.6m2 2.3m2)
• Example 2 What is the length of side B if side A
  is 4.3 m and side C is 10.5 m?
• Answer
• 1. Given side A 4.3m, side C 10.5m
• 2. Looking for length of side B
• 3. Equation B v(C2 A2)
• 4. Solution B v(C2 A2) v(10.5m2 4.3m2)
• Example 3 What is the length of side C if side A
  is 8.7 m and side B is 3.1 m?
• Answer
• 1. Given side A 8.7m, side B 3.1m
• 2. Looking for length of side C
• 3. Equation C v(A2 B2)
• 4. Solution C v(A2 B2) v(8.7m2 3.1m2)

13
Acceleration

• An object that has an acceleration is one whose
  velocity is changing
• The change in velocity can be one of three
  things
• An increase in speed (i.e. going faster)
• A decrease in speed (i.e. going slower)
• A change in direction of the speed (i.e. turning)
• Acceleration is a vector quantity
• When acceleration is in a straight line
• Acceleration change in speed / time interval
• OR
• a (v2 - v1) / t, where a is the acceleration,
  v2 is the final speed, v1 is the initial speed,
  and t is the time interval
• Common units to express acceleration are m/s2
• When acceleration is from a decrease in speed, it
  is called negative acceleration or deceleration
• We will assume that acceleration is constant,
  although that is not always the case

14
Example Calculations With Acceleration

• Example If an objects initial speed is 30m/s
  and its final speed is 55m/s, what is its
  acceleration over 571 seconds?
• Answer
• 1. Given initial speed 30m/s, final speed
  55m/s, 571 seconds
• 2. Looking for acceleration
• 3. Equation a (v2 - v1) / t
• 4. Solution a (v2 - v1) / t (55m/s 30m/s)
  / 571 s 0.0438 m/s2

15
Using Acceleration to Calculate Final Speed

• To calculate the final speed of something, the
  equation for acceleration (a (v2 - v1) / t )
  needs to be rewritten
• 1. First bring t over to the other side of the
  equation by multiplying it on both sides
• 2. Then, separate v2 from v1 by adding v1 to both
  sides
• The equation to calculate final speed is v2 v1
  at

• Example If an object has an initial speed of 60
  m/s with an acceleration of 3.2m/s2, what is its
  final speed after 43 seconds?
• Answer
• 1. Given initial speed 60m/s, acceleration
  3.2m/s2, 43 seconds
• 2. Looking for final speed
• 3. Equation v2 v1 at
• 4. Solution v2 v1 at 60m/s
  (3.2m/s2)(43s) 197.6m/s

16
How Far?

• To calculate the average speed of an object
  during uniform acceleration
• average speed (v1 v2) / 2
• The equation to calculate the distance the object
  moves is d vt
• If the speed, v, is an average speed, then the
  equation for distance is d (average speed)t
• If we insert the equation for the average speed
  into this equation, then the equation for
  distance is d (v1 v2)/2t (v1t/2)
  (v2t/2)
• If we insert the equation for the final speed
  into this equation, then the equation for
  distance is d (v1t/2) ((v1 at)t/2)
  (v1t/2) (v1t/2) (at2/2) v1t (at2/2)
• So, d v1t (at2/2)
• If the object starts at rest, then v1 0, so d
  (at2/2)

17
Example Calculations for Average Speed and
Distance

• Average Speed Example What is the average speed
  of an object that was moving at some times 45m/s
  and at other times 35m/s?
• Answer
• 1. Given speed 1 45m/s, speed 2 35m/s
• 2. Looking for average speed
• 3. Equation average speed (v1 v2) / 2
• 4. Solution average speed (v1 v2) / 2
  (45m/s 35m/s)/2 40m/s

• Distance Example 1 If an object has an initial
  speed of 10 m/s with an acceleration of 1.5m/s2,
  how far did it travel in 76 seconds?
• Answer
• 1. Given initial speed 10m/s, acceleration
  1.5m/s2, 76 seconds
• 2. Looking for how far distance
• 3. Equation d v1t (at2/2)
• 4. Solution d v1t (at2/2) (10m/s)(76s)
  (1.5m/s2)(76s)2/2 5092m
• Distance Example 2 If an object starts from rest
  with an acceleration of 0.45m/s2, how far did it
  travel in 5.6 seconds?
• Answer
• 1. Given acceleration 0.45m/s2, 5.6 seconds,
  initial speed 0 m/s (at rest)
• 2. Looking for how far distance
• 3. Equation d (at2/2)
• 4. Solution d (at2/2) (0.45m/s2)(5.6s)2/2
  7.06m

18
Acceleration of Gravity

• Before Galileo, philosophers tried to answer
  questions about the movement of objects due to
  gravity by creating concepts that were so obvious
  that there was no need to test them
• Aristotle, an ancient Greek thinker, was one of
  these famous philosophers
• He thought that all falling bodies followed this
  basic concept every material has a natural
  place where it belonged and toward where it tried
  to move
• For example, stones are from the earth, so they
  naturally fell downward, toward the earth

19
Free Fall

• Galileo, an Italian physicist, experimented with
  falling bodies 2000 years after Aristotle
• He performed his experiments using objects
  rolling down an inclined plane instead of falling
  in free fall, but his results apply to objects in
  free fall as well
• He found that the higher an object is dropped,
  the greater its speed when it reaches the ground,
  meaning that the object has an acceleration
• He also found that no matter the size of the
  object, whether it is big or small, the
  acceleration is the same
• Galileo found that if an object that is falling
  near the earths surface doesnt have to push its
  way through any air, it has an acceleration of
  9.8 m/s2 this acceleration is known as the
  acceleration of gravity, and is represented by g
• When an object is falling from rest, its downward
  speed can be calculated by the following
  equation vdownward gt, where g is 9.8 m/s2 and
  t is the time

20
How Far Does a Falling Object Fall?

• To determine how far (h) a falling object falls
  from rest in a given time
• 1. Start with d at2/2
• 2. If you replace d, distance, with h, height,
  the equation is h at2/2
• 3. If you replace a, acceleration, with g, the
  acceleration of gravity, the equation is h
  gt2/2

21
A Falling Object Speed vs. Height

• The downward speed is calculated by the following
  equation vdownward gt
• This equation shows that the speed is directly
  proportional to the time i.e., the speed at 5
  seconds is 5 times the speed at 1 second
• The height is calculated by the following
  equation h gt2/2
• This equation shows that the height is directly
  proportional to the time squared (t2) - i.e., the
  speed at 5 seconds is 25 times the speed at 1
  second
• The height thus increases faster with increasing
  time than the downward speed

22
Example Calculations of Downward Speed and Height

• Downward Speed Example What is the speed of an
  object that is falling for 34 seconds?
• Answer
• 1. Given 34 seconds, a g
• 2. Looking for speed
• 3. Equation vdownward gt
• 4. Solution vdownward gt (9.8m/s2)(34s)
  333.2m/s

• Height Example How far does an object fall after
  4.9 seconds?
• Answer
• 1. Given 4.9 seconds, a g
• 2. Looking for how far height
• 3. Equation h gt2/2
• 4. Solution h gt2/2 (9.8m/s2)(4.9s)2/2
  117.6m

23
Thrown Objects

• The acceleration of gravity, g, is the same
  whether an object is
• just dropped
• thrown downward
• thrown upward
• thrown sideways
• If a ball is just dropped, it goes faster and
  faster as it drops until it hits the ground
• If a ball is thrown sideways, it has a curved
  path that becomes steeper and steeper as it drops
• If a ball is thrown upward, gravity first reduces
  the upward speed of the ball until its upward
  speed is zero. When its upward speed is zero, the
  ball is at the top of its path and is momentarily
  at rest. Then the ball starts falling toward the
  ground, faster and faster, until it hits the
  ground.
• If a ball is thrown downward, the original speed
  of the ball is steadily increased by the downward
  acceleration (g)

24
Parabolas

• A parabola is a curved path
• When a ball is thrown upward at an angle to the
  ground, a parabola occurs
• If there is no air resistance, the maximum
  distance a ball can travel horizontally occurs
  when the ball is thrown upward at an angle of 45
  to the ground
• If the angle is greater than 45 or less than
  45, then the ball travels less than it would
  have if it had be thrown at an angle of 45
• There is one angle above 45 and one angle below
  45 that if the ball is thrown, it will land in
  the same place

25
Air Resistance

• Air resistance keeps things from developing the
  full acceleration of gravity without air
  resistance, anything, including a light shower,
  would be dangerous
• The faster something moves, the more air
  resistance it encounters - i.e. the more the air
  slows down its motion
• For a falling object, the air resistance
  increases with speed until the air resistance
  equals the force of gravity on the object. Once
  this occurs, the object falls at a constant
  terminal speed. The terminal speed of an object
  depends on its size, its shape, and how heavy it
  is

26
Air Resistance for Objects Thrown Upward at an
Angle

• If there is no air resistance (i.e., in a
  vacuum), the maximum distance a ball can travel
  horizontally occurs when the ball is thrown
  upward at an angle of 45 to the ground
• However, if there is air resistance, the maximum
  distance a ball can travel horizontally doesnt
  occur when the ball is thrown upward at an angle
  of 45 to the ground it occurs when the ball is
  thrown upward at an angle less than 45 to the
  ground

27
Falling Air vs. Vacuum

• In air, a stone falls faster than a feather
  because the air resistance affects the feather
  more than the stone
• In a vacuum, since there is no air resistance,
  both a stone and a feather fall with the same
  acceleration, g (9.8 m/s2)