representing motion chapter 2. some definitions and assumptions about motion particle model : focus...
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Representing MotionChapter 2
Some Definitions and Assumptions about Motion
Particle model: focus on a single point on an extended object; the size of the object must be less than the distance it moves.
Use a coordinate system to describe an object’s position, i.e., the separation between where the object is and the origin of the coordinate system. We can locate the origin in a convenient place.
In describing motion we’ll use two different quantities.
Scalars have a single attribute, a magnitude.
Examples: distance, speed, time, temperature
Vectors have two attributes, a magnitude and a direction.
Examples: displacement, velocity, acceleration, force
Representing Vectors in One Dimension
0 1 2 3 4-1-2-3-4
AB
• Vector A has a magnitude of 3 units and is in the +x direction
• Vector B has a magnitude of 2 units and is in the –x direction
• We can also represent vectors as a symbol with an arrow over it—e.g., means the vector A.
x
Combining Vectors in One Dimension
0 1 2 3 4-1-2-3-4
A
• Vector A has a magnitude of 3 units and is in the +x direction• Vector B has a magnitude of 2 units and is in the +x direction• If C = A + B, the resultant vector C has a magnitude of 5 units and
is in the +x direction.• If D = A – B, the resultant vector D has a magnitude of 1 unit and is
in the +x direction (because A – B = A + (-B) and the vector –B has the same magnitude as B and points in the opposite direction).
• We can also multiply vectors by scalars. The vector E = nA is a vector whose magnitude is n times the magnitude of A and points in the same direction as A. So if n=3 then the magnitude of E is 9 units and it is in the +x direction.
x
5
B
C
D
Exercises with Vectors in 1-D
Suppose we’re given three vectors
A is 5 units in magnitude and points in the +x direction
B is 3 units in magnitude and points in the –x direction
C is 2 units in magnitude and points in the +x direction Describe (magnitude and direction) the following
resultant vectors W, X, Y and Z where
1.W = A – B
2.X = A + B + C
3.Y = B + nA; where n=4
4.Z = A/5
Displacement and Time Intervals In analyzing motion we’ll want to know how long it takes an
object to move from one position to another.
We’ll define the displacement to be the change in the object’s position.
where is the initial position and is the final position.
Note that displacement is a vector that starts at the object’s initial position and ends at the object’s final position
Similarly we’ll define the time interval to be the time it takes for the object to move from its initial position to its final position.
Difference between Distance and Displacement
video
Velocity and Speed Define the average velocity to be the change in position (i.e.,
displacement) divided by the time during which the change occurred , i.e.,
Note that is a vector.
The average speed is the magnitude of the average velocity (so it’s a scalar).
The instantaneous velocity is the speed and direction of an object at a particular instant.
Note that the units of velocity and speed are e.g., or , etc.
Equation of Motion for Object Moving with Average Velocity
We can choose to start the clock at and let . Then
That is, the final position of an object is equal to its average velocity multiplied by the time, plus its initial position.
Note this has the same form as the equation of a straight line
.
So if we plot vs. for various values of , the slope of the line will be the average velocity and the y-intercept will be the initial position .
Plotting Position vs. Time
0 5 10 15 20 25 30 350
10
20
30
40
50
60
70
Rise = 35
Run = 17.5
Slope=𝑅𝑖𝑠𝑒𝑅𝑢𝑛
=3517.5
=2Po
siti
on
Time
y-intercept
From the graph the average velocity is 2 (in units of length/time) and theinitial position is 5 (in units of length). So the equation that describes the position of the object as a function of time is .
2009 World Championships in Berlin
Usain Bolt 2009 World Record
Usain Bolt’s 100m World RecordDistance (m) Time (s) Split (s) 10m Average
Velocity (m/s)
Reaction Time 0.146
10 1.85 1.704 5.87
20 2.89 1.04 9.62
30 3.78 0.89 11.24
40 4.64 0.86 11.63
50 5.49 0.85 11.76
60 6.31 0.82 12.20
70 7.11 0.80 12.50
80 7.92 0.81 12.35
90 8.74 0.82 12.20
100 9.58 0.84 11.90
Bolt’s Position vs. Time Graph
0.00 2.00 4.00 6.00 8.00 10.00 12.000
20
40
60
80
100
120
Time (s)
Posit
ion
(m
)
How Fast are Humans?
Bolt’s fastest split was between 60 and 70 meters; his average velocity during that split was 12.50 m/s.
Convert that velocity to mi/hr:
Top speeds of various animals
Cheetah – 75 mi/hr
Greyhound – 40 mi/hr
Domestic cat – 30 mi/hr
Brown bear – 22 mi/hr
How Fast is 0.01 Second?
Usain Bolt’s time: 9.79s; Justin Gatlin’s time: 9.80sEstimate how far apart in distance they were at the finish.
Usain Bolt 2015 World Championship
Problem
D 100m
• At the 2015 NJ Outdoor State Championships Charlie Volker from Rumson-Fair Haven HS won the Group 2 100m dash race with a time of 10.97 seconds.
• How long of a lead (D) could Usain Bolt give him and finish the race in the same time?
UB
CV
Activities
Super Ultimate Graphing Challenge
Boat Motion Lab
New Horizons Mission to Pluto NASA’s New Horizons spacecraft was
launched on January 19, 2006 and reached Pluto on July 14, 2015. So, it took approx. 9 years and 6 months to reach to Pluto.
The spacecraft traveled a total distance of 39 Astronomical Units (AU) during that time. (1 AU is the average distance between the Earth and the sun and is equal to 93 million miles.)
What was the average speed of the New Horizons spacecraft in AU/year and mi/hr?
Radio communications travel at the speed of light (approx. or ). How much time does it take a radio signal from the spacecraft to reach the Earth when the spacecraft is near Pluto?