Understanding Motion Along a Straight Line

 
Unit: Motion Along A
Straight Line
 
 
Unit: Motion Along A Straight Line
 
How do you describe the motion
of a particle?
 
Why can a woodpecker survive
the severe impacts with a tree
limb?
 
 
Chapter 2
Motion Along a Straight Line
 
In this chapter we will study 
kinematics
, i.e., how objects move along a 
straight
line.
The following parameters will be defined:
Displacement
Average velocity
Average speed
Instantaneous velocity
Average and instantaneous acceleration
For constant acceleration we will develop the equations that give us the velocity
and position at any time. In particular we will study the motion under the
influence of gravity close to the surface of the Earth.
Finally, we will study a graphical integration method that can be used to analyze
the motion when the acceleration is not constant.
 
Kinematics vs. Dynamics
 
• Kinematics – is the study how things move
 
 
• Dynamics – is the study of why things move
 
Kinematics
 is the part of mechanics that describes the motion of physical
objects. We say that an object moves when its position as determined by an
observer changes with time.
In this chapter we will study a restricted class of kinematics problems
Motion will be along a straight line.
We will assume that the moving objects are 
“particles
,” i.e., we restrict our
discussion to the motion of objects for which all the points move in the same
way.
 The causes of the motion will not be investigated. This will be done
later in the course.
Consider an object moving along a straight line 
taken
to be the x-axis. The object’s position at any time t is
described by its coordinate x(t) defined with respect to
the origin O. The coordinate x can be positive or
negative depending whether the object is located
on the positive or the negative part of (2-2) the x-axis.
 
Displacement
 
Displacement.
If an object moves from position         to position      , the change in position is described by the
displacement
 
 For example if                 and              then ∆x = 12 – 5 = 7 m.
The positive sign of ∆x indicates that the motion is along the positive x-direction.
 If instead the object moves from                    and                 then ∆x = 1 – 5 = -4 m.
The negative sign of ∆x indicates that the motion is along the negative x direction. Displacement is a
vector quantity that has both magnitude and direction. In this restricted one-dimensional motion
the direction is described by the algebraic sign of ∆x.
Note: The actual distance for a trip is irrelevant as far as the
 displacement is concerned.
Consider as an example the motion of an object from
 an initial position            m to x = 200 m and then back to             .
Even though the total 
distance covered is 390 m the displacement
then is ∆x = 0.
 
x (m)
t (s)
14
What is the average velocity of a particle, whose motion
is described by the below position - time graph, from
t = 0 s to t = 4 s?
A
-2 m/s
B
-0.5 m/s
C
0 m/s
D
+0.5 m/s
E
+2.0 m/s
https://njctl.org/video/?v=ygGfndEMwqk
x (m)
t (s)
15
What is the average velocity of a particle, whose motion
is described by the below position - time graph, from
t = 2 s to t = 4 s?
A
+3.0 m/s
B
0.33 m/s
C
0 m/s
D
-0.33 m/s
E
-3.0 m/s
https://njctl.org/video/?v=SUnZAgecI6Y
x (m)
t (s)
16
Explain, using the definition of slope, why the average
velocity from t = 0 s to t = 4 s was less than the average
velocity from t = 2 s to t = 4 s.
https://njctl.org/video/?v=x7dcBMXySbM
 
Derivatives
 
Looking at these two graphs, which one can you find the instantanious velocity at 15
seconds and which one can you not? Also, explain why you can’t find the instantaneous
velocity for that graph.
 
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Average velocity is used to define a particle's motion over a specified
interval of time, but that is not always what is most interesting.  For
example, a police officer is not interested in your average velocity over
a two hour period.  But, he is very interested in your velocity at a
specific given time.
 
This is more fundamental then it seems - the officer, even though he
probably doesn't realize it, actually wants your velocity at a time
interval of zero seconds!
 
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Let's see how the average velocity equation works for a zero second
time interval.  It's problematic because algebra doesn't handle
fractions with a zero in the denominator.
 
 
 
Sir Isaac Newton recognized this problem and came up with the
concept of a zero time interval and invented calculus so he could solve
problems like this.
 
Calculus involves taking a time interval and shrinking it until it is
infinitesimally close to zero.  You'll learn more about this in your
Calculus class - but you can tell the teacher you first saw it here.
Calculus was invented for Physics.
 
 
Δt = 0 is not good
 
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P
2
 
P
1
 
Δ
x
 
Δ
t
 
t (s)
 
x (m)
 
P
2
 
P
1
 
Δ
x
 
Δ
t
 
x (m)
 
t (s)
 
Point P
2
 moved closer to the fixed point P
1
 as the time interval
between positions was made smaller.  Note that Δx decreased.
The slope of the line connecting the two points (the average
velocity) also changed.  What should we continue doing?
 
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Continue to decrease Δt, until P
2
 is infinitesimally close to P
1
 - just as
Δt gets infinitesimally close to zero.  When we get "there," the slope of
the line at P
1
 is the instantaneous velocity.  It is also tangent to the
position - time curve at time t
1
.
 
P
1
 
t (s)
 
x (m)
 
Slope of tangent = instantaneous velocity
 
t
1
 
t
2
 
x
2
 
x
1
 
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The average velocity equation was used as Δt kept decreasing until it
almost hit zero, at which point the instantaneous velocity of the
particle was found.  We call that a "limit," as follows:
 
 
 
 
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The derivative of a graphed function is also the slope of the tangent
line to the function at that point.
 
The derivative of the position is the velocity.
 
Your calculus course will cover this in more depth.
17
Explain how instantaneous velocity is derived by starting
with average velocity.  Use algebra, graphical analysis
and a little bit of calculus in your response.
https://njctl.org/video/?v=R8mEo3ABlOM
 
Finding the Derivative (Power Rule)
 
 
 
 
Example:
Example: 20
Derive the
following
function
f(x) = 3x - 2
 
f’(x) = 3
Example: 21
Derive the
following function
f(x) = -x
2
 +4x + 5
 
f’(x) = -2x + 4
Example: 22
Derive the
following function
f(x) = -x
2
 + 12x - 28
 
f’(x) =-2x +12
21
If the position of a particle as a function of time is
x(t) = 2t
3
 + 4t
2
 + t + 18, what is its velocity at t = 3 s?
A
 61 m/s
B
 79 m/s
C
 97 m/s
D
103 m/s
E
111 m/s
https://njctl.org/video/?v=j-gh0HCSLi8
 
Integration
 
Looking at these two graphs, which one can you find the total displacement? Also, explain why
you can’t find the displacement for the other graph.
 
Finding the Integral Power Rule
 
 
 
 
 
 
Example
 
 
If        is continuous on 
[a,b] 
then the area under the curve is
the integral of        from 
a
 to 
b
.
 
The integral represents the "net area" meaning all area above
the 
x
-axis minus any area below the 
x
-axis.
 
The expression is read as:
"The integral from
 
a
 
to
 
b
 
of 
f
(
x
)
 dx
."
or
"The integral from 
a
 
to
 
b
 
of 
f
(
x
)
 with respect to 
x
."
 
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There are many ways to describe an antiderivative, but the
most common is to say it is the "opposite of taking the
derivative". In other words, if you are given the derivative of a
function, the antiderivative would be the original function that
the derivative came from.
 
Antiderivative
: A function,   , is an antiderivative of the
function    if              .
 
if and only if
 
Therefore,
5
Evaluate the following integral:
A
24
B
44
C
34
D
38
E
I need help
https://njctl.org/video/?v=AZb_udFWgmo
 
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This is a Quick Reference for common Antiderivatives.
 
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The "
C
" represents a constant term in the antiderivative. We do not
know what it is because when finding a derivative initially, the
constant term in the function "disappears" or rather
 
 
 
When evaluating definite integrals, we do not need to worry about it,
because it would cancel out (regardless of its value) when you
subtract like this:
6
Evaluate:
A
4
B
−8
C
8
D
−4
E
I need help
 
https://njctl.org/video/?v=dSvCPsQScNQ
Example 28
 
104 m
An object starts at an initial position of 0m. Its velocity as a
function of time is v(t) = 3t
2
 + 2t + 6. What is its position at t = 4s.
Conceptual Example #4
(A) Under which scenario does the car’s
speed increase? Decrease?
(B) In which scenario does the car have a
negative acceleration?
 
A & D Increase
B & C Decrease
 
B & D
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Exploring kinematics in the context of motion along a straight line, this chapter delves into parameters like displacement, velocity, speed, and acceleration. Kinematics is distinguished from dynamics, focusing on how objects move without delving into why. Displacement, a key concept, is discussed as a vector quantity with both magnitude and direction. Examples illustrate how displacement is independent of actual distance traveled. The chapter sets the foundation for studying motion in one dimension.


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  1. Unit: Motion Along A Straight Line

  2. Unit: Motion Along A Straight Line How do you describe the motion of a particle? Why can a woodpecker survive the severe impacts with a tree limb?

  3. Chapter 2 Motion Along a Straight Line In this chapter we will study kinematics, i.e., how objects move along a straight line. The following parameters will be defined: Displacement Average velocity Average speed Instantaneous velocity Average and instantaneous acceleration For constant acceleration we will develop the equations that give us the velocity and position at any time. In particular we will study the motion under the influence of gravity close to the surface of the Earth. Finally, we will study a graphical integration method that can be used to analyze the motion when the acceleration is not constant.

  4. Kinematics vs. Dynamics Kinematics is the study how things move Dynamics is the study of why things move

  5. Kinematics is the part of mechanics that describes the motion of physical objects. We say that an object moves when its position as determined by an observer changes with time. In this chapter we will study a restricted class of kinematics problems Motion will be along a straight line. We will assume that the moving objects are particles, i.e., we restrict our discussion to the motion of objects for which all the points move in the same way. The causes of the motion will not be investigated. This will be done later in the course. Consider an object moving along a straight line taken to be the x-axis. The object s position at any time t is described by its coordinate x(t) defined with respect to the origin O. The coordinate x can be positive or negative depending whether the object is located on the positive or the negative part of (2-2) the x-axis.

  6. Displacement Displacement. If an object moves from position to position , the change in position is described by the displacement 1x 2x = For example if and then x = 12 5 = 7 m. The positive sign of x indicates that the motion is along the positive x-direction. If instead the object moves from and then x = 1 5 = -4 m. The negative sign of x indicates that the motion is along the negative x direction. Displacement is a vector quantity that has both magnitude and direction. In this restricted one-dimensional motion the direction is described by the algebraic sign of x. Note: The actual distance for a trip is irrelevant as far as the displacement is concerned. Consider as an example the motion of an object from an initial position m to x = 200 m and then back to . Even though the total distance covered is 390 m the displacement then is x = 0. 5 x m = 12 x m 1 2 = 5 x m = 1 x m 1 2 = 5 x m = 5 x m 1 2

  7. 14 What is the average velocity of a particle, whose motion is described by the below position - time graph, from t = 0 s to t = 4 s? x (m) A -2 m/s B -0.5 m/s C 0 m/s D +0.5 m/s E +2.0 m/s Answer E t (s) https://njctl.org/video/?v=ygGfndEMwqk

  8. 15 What is the average velocity of a particle, whose motion is described by the below position - time graph, from t = 2 s to t = 4 s? x (m) A +3.0 m/s B 0.33 m/s C 0 m/s D -0.33 m/s E -3.0 m/s Answer A t (s) https://njctl.org/video/?v=SUnZAgecI6Y

  9. 16 Explain, using the definition of slope, why the average velocity from t = 0 s to t = 4 s was less than the average velocity from t = 2 s to t = 4 s. x (m) The slope of the line connecting the particle's position change between 0 and 4 s was less than the slope of the line connecting the particle's position change from 2 to 4 s. Hence, the average velocity was greater for the 2 to 4 s interval. Answer t (s) https://njctl.org/video/?v=x7dcBMXySbM

  10. Derivatives Looking at these two graphs, which one can you find the instantanious velocity at 15 seconds and which one can you not? Also, explain why you can t find the instantaneous velocity for that graph.

  11. Instantaneous Velocity Average velocity is used to define a particle's motion over a specified interval of time, but that is not always what is most interesting. For example, a police officer is not interested in your average velocity over a two hour period. But, he is very interested in your velocity at a specific given time. This is more fundamental then it seems - the officer, even though he probably doesn't realize it, actually wants your velocity at a time interval of zero seconds! The officer is looking for your instantaneous velocity to ensure you're obeying the speed limit. https://njctl.org/video/?v=TkFXKb--Q8s

  12. Instantaneous Velocity Let's see how the average velocity equation works for a zero second time interval. It's problematic because algebra doesn't handle fractions with a zero in the denominator. t = 0 is not good Sir Isaac Newton recognized this problem and came up with the concept of a zero time interval and invented calculus so he could solve problems like this. Calculus involves taking a time interval and shrinking it until it is infinitesimally close to zero. You'll learn more about this in your Calculus class - but you can tell the teacher you first saw it here. Calculus was invented for Physics.

  13. Instantaneous Velocity Point P2 moved closer to the fixed point P1 as the time interval between positions was made smaller. Note that x decreased. The slope of the line connecting the two points (the average velocity) also changed. What should we continue doing? x (m) x (m) P2 P2 x x P1 P1 t t (s) t (s) t

  14. Instantaneous Velocity Continue to decrease t, until P2 is infinitesimally close to P1 - just as t gets infinitesimally close to zero. When we get "there," the slope of the line at P1 is the instantaneous velocity. It is also tangent to the position - time curve at time t1. x (m) x2 P1 x1 t (s) t2 t1

  15. Instantaneous Velocity The average velocity equation was used as t kept decreasing until it almost hit zero, at which point the instantaneous velocity of the particle was found. We call that a "limit," as follows: Instantaneous velocity is labeled vx, and the dx/dt term is called the "derivative of x with respect to t." The derivative of a graphed function is also the slope of the tangent line to the function at that point. The derivative of the position is the velocity. Your calculus course will cover this in more depth.

  16. 17 Explain how instantaneous velocity is derived by starting with average velocity. Use algebra, graphical analysis and a little bit of calculus in your response. Use the definition for average velocity, but then keep shrinking the time interval; this will also decrease the change in position. When the time interval is close to zero, average velocity is now the instantaneous velocity. The slope of the tangent line at the point of interest is the instantaneous velocity. Answer https://njctl.org/video/?v=R8mEo3ABlOM

  17. Finding the Derivative (Power Rule) Example:

  18. Example: 20 Derive the following function f (x) = 3 f(x) = 3x - 2

  19. Example: 21 Derive the following function f(x) = -x2 +4x + 5 f (x) = -2x + 4

  20. Example: 22 Derive the following function f(x) = -x2 + 12x - 28 f (x) =-2x +12

  21. 21 If the position of a particle as a function of time is x(t) = 2t3 + 4t2 + t + 18, what is its velocity at t = 3 s? A B C D 103 m/s E 111 m/s 61 m/s 79 m/s 97 m/s Answer B https://njctl.org/video/?v=j-gh0HCSLi8

  22. Integration Looking at these two graphs, which one can you find the total displacement? Also, explain why you can t find the displacement for the other graph.

  23. Finding the Integral Power Rule Example

  24. The Definite Integral If is continuous on [a,b] then the area under the curve is the integral of from a to b. The integral represents the "net area" meaning all area above the x-axis minus any area below the x-axis. The expression is read as: "The integral fromatobof f(x) dx." or "The integral from atobof f(x) with respect to x."

  25. The Antiderivative There are many ways to describe an antiderivative, but the most common is to say it is the "opposite of taking the derivative". In other words, if you are given the derivative of a function, the antiderivative would be the original function that the derivative came from. Antiderivative: A function, , is an antiderivative of the function if . Therefore, if and only if

  26. 5 Evaluate the following integral: A 24 C Answer B 44 antiderivative C 34 D 38 E I need help https://njctl.org/video/?v=AZb_udFWgmo

  27. Calculating Antiderivatives This is a Quick Reference for common Antiderivatives.

  28. Calculating Antiderivatives The "C" represents a constant term in the antiderivative. We do not know what it is because when finding a derivative initially, the constant term in the function "disappears" or rather When evaluating definite integrals, we do not need to worry about it, because it would cancel out (regardless of its value) when you subtract like this:

  29. 6 Evaluate: A 4 A B 8 Answer C 8 D 4 E I need help https://njctl.org/video/?v=dSvCPsQScNQ

  30. Example 28 An object starts at an initial position of 0m. Its velocity as a function of time is v(t) = 3t2 + 2t + 6. What is its position at t = 4s. 104 m

  31. Conceptual Example #4 (A) Under which scenario does the car s speed increase? Decrease? (B) In which scenario does the car have a negative acceleration? A & D Increase B & C Decrease B & D

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