Relationships Among Kinematics, Newton's Laws, Vectors, 2D Motion, 2D Forces, Momentum, Work, Energy, and Powerhttp://cnx.org/contenthttp://cnx.org/content/m38369/latest/m38369Relationships Among Kinematics, Newton's Laws, Vectors, 2D Motion, 2D Forces, Momentum, Work, Energy, and Power1.22011/05/17 19:05:29 -05002012/10/08 14:03:36.846 GMT-5RichardBaldwinRichard BaldwinBaldwin@DickBaldwin.combaldwinbaldwinbaldwinaccessibilityaccessibleblindconservative forceelastic potential energyexternal forcegraph boardgravitational potential energyinternal forceJavaScriptkinetic energymechanical energynon-conservative forcephysicspotential energypowerprotractorrefreshable Braille displayscreen readertotal mechanical energytrigonometrywattworkScience and TechnologyThis module illustrates relationships among kinematics, Newton's laws, vectors, 2D motion, 2D forces, momentum, work, energy, and power in a format that is accessible to blind students.en
Table of Contents
Preface
General
Prerequisites
Supplemental material
Discussion
An ideal rocket example
Leg A
Leg B
Leg C
Leg D
Do the calculations
Resources
Miscellaneous
<emphasis id="Preface" effect="bold">
</emphasis>
Preface
<emphasis id="General" effect="bold">
General
</emphasis>
This module is part of a collection (see
http://cnx.org/content/col11294/latest/
) of modules designed to make physics
concepts accessible to blind students. The collection is intended to supplement but not to
replace the textbook in an introductory course in high school or college
physics.
This module illustrates relationships among kinematics, Newton's laws,
vectors, 2D linear motion, 2D forces, momentum, work, energy, and power in a
format that is accessible to blind students.
<emphasis id="Prerequisites" effect="bold">
Prerequisites
</emphasis>
In addition to an Internet connection and a browser, you will need the
following tools (as a minimum) to work through the exercises in these modules:
A graph board for plotting graphs and vector diagrams (
http://www.youtube.com/watch?v=c8plj9UsJbg
).
A protractor for measuring angles (
http://www.youtube.com/watch?v=v-F06HgiUpw
).
An audio screen reader that is compatible with your operating system,
such as the NonVisual Desktop Access program (NVDA), which is freely
available at
http://www.nvda-project.org/
.
A refreshable Braille display capable of providing a line by line tactile output of information displayed on the computer monitor
(
http://www.userite.com/ecampus/lesson1/tools.php
).
A device to create Braille labels. Will be used to label graphs
constructed on the graph board.
The minimum prerequisites for understanding the material in these modules
include:
A good understanding of algebra.
An understanding of the use of a graph board for plotting graphs and
vector diagrams (
http://www.youtube.com/watch?v=c8plj9UsJbg
).
An understanding of the use of a protractor for measuring angles (
http://www.youtube.com/watch?v=v-F06HgiUpw
).
A basic understanding of the use of sine, cosine, and tangent from
trigonometry (
http://www.clarku.edu/~djoyce/trig/
).
An introductory understanding of JavaScript programming (
http://www.dickbaldwin.com/tocjscript1.htm
and
http://www.w3schools.com/js/default.asp
).
An understanding of all of the material covered in the earlier modules
in this collection.
<emphasis id="Supplemental_material" effect="bold">
Supplemental material
</emphasis>
I recommend that you also study the other lessons in my extensive collection
of online programming tutorials. You will find a consolidated index at
www.DickBaldwin.com
.
<emphasis id="General_background_information" effect="bold">
Discussion
</emphasis>
A wrap-up module
The next module following this one will involve circular motion, which will
be a major change in direction (no pun intended). Therefore, in this module, I
will work through a major example involving a rocket that will tie together much
of what you have learned in earlier modules.
Before we get to that example, however, let's do a quick review on external and
internal forces.
A quick review of external forces
You learned in an earlier module that when work is done on an object by
external
forces, the total mechanical energy
possessed by the object, consisting of kinetic energy plus potential energy,
must change.
The work done on the object by external forces can be positive, in which case the total mechanical energy will increase. The work can be negative, in which case the total mechanical energy will decrease. The change in mechanical energy will be equal to the net work that is done on the object.
In this case, the total mechanical energy is not
conserved
.
Therefore, external forces are often referred to as
non-conservative
forces.
A quick review of internal forces
On the other hand, you also learned that if work is done on an object only by
internal
forces, the total mechanical energy possessed by the object
cannot change. However, it can be transformed from potential energy to kinetic
energy and vice versa.
In this case, the total mechanical energy is
conserved
. Therefore,
internal forces are often referred to as
conservative
forces.
A quantitative relationship
The quantitative relationship between work and mechanical energy can be
stated as follows:
MEf = MEi + We
where
MEf and MEi represent the final and initial total mechanical energy
possessed by the object respectively.
We represents the work done on the object by external forces.
This equation states that the final amount of mechanical energy possessed by
an object is equal to the initial mechanical energy plus the work done on the
object by external forces.
Potential energy plus kinetic energy
The total mechanical energy at any point in time can be the sum of potential
energy
(gravitational or elastic potential energy)
and kinetic energy
due to motion.
Given that, we can rewrite the earlier equation as:
KEf + PEf = KEi + PEi + We
where
KEf and PEf represent the final kinetic and potential energy
respectively.
KEi and PEi represent the initial kinetic and potential energy
respectively.
We represents the work done on the object by external forces.
As mentioned, the work done by external forces can be either positive or
negative work. Whether the work is positive or negative depends on the cosine of
the angle between the direction of the force and the direction of the
displacement of the object.
<emphasis id="An_ideal_rocket_example" effect="bold">
An ideal rocket example
</emphasis>
Consider the following scenario. The owners of an experimental rocket lift the rocket
onto a platform above ground level and set it up for firing.
Later, when they fire the
rocket, it goes straight up while the rocket engine is burning. When the rocket
engine runs
out of fuel and stops burning, the rocket coasts to its apex and stops climbing.
Then it falls back to the surface of the earth in an unglamorous free fall.
Simplifying assumptions
We will make some simplifying assumptions:
The mass of the fuel is insignificant relative to the combined mass of
the rocket and its payload. Therefore, expenditure of fuel doesn't affect
the mass of the rocket in a significant way.
Air resistance is negligible. The rocket acts as if in a vacuum.
Initial conditions
Here are the initial conditions for the rocket experiment:
Platform height = 15 meters.
Mass of rocket and payload = 10kg.
Thrust of rocket is constant at 150 newtons during burn.
Burn time for the rocket = 10 seconds.
Legs of the trip
We will analyze the rocket's round trip from the ground, into the air, and
back to the ground in several legs as described below:
Leg A: Manually lifting the rocket from the ground to the platform.
Leg B: Displacement of the rocket under rocket-engine power straight up.
Leg C: Displacement of the rocket without power while coasting to the apex.
Leg D: Displacement of the rocket in free fall from the apex back to the
ground.
We will analyze several aspects of the state of the rocket at the end of each
leg. We will also compare alternative ways of computing the state of the rocket.
<emphasis id="Leg_A" effect="bold">
Leg A
</emphasis>
During this leg, the rocket is manually lifted from the ground to the platform.
The rocket has
no potential or kinetic energy while on the ground, so it begins with zero
mechanical energy.
An external force must be provided to lift the rocket from the ground to the
platform in order to overcome the internal force of gravity. As an external
force, this force is capable of changing its mechanical energy, which it does.
When the rocket has been lifted onto the platform, the mechanical energy of
the rocket consists of its gravitational potential energy, which is equal to
the work done to lift it to the platform. The kinetic energy will be 0 at that
point because the rocket isn't moving.
Weight of the rocket = m*g = 10kg*9.8m/s^2 = 98 newtons
Work = f * d = 98N * 15m = 1470 joules
State at the end of Leg A
Thus, the total mechanical energy possessed by the rocket at the end of Leg A is 1470 joules.
<emphasis id="Leg_B" effect="bold">
Leg B
</emphasis>
During this leg, which begins when the rocket engine fires, the rocket flies straight up as a result of a constant
upward force exerted by the rocket engine.
The net acceleration
For this leg, we need to determine
the net acceleration that is applied to the rocket. The net acceleration
consists of the upward or positive acceleration due to the force of the rocket
engine and the downward or negative acceleration of gravity.
Aup = 150N/10kg = 15 m/s^2
Ag = -9.8 m/s^2
Anet = 15 m/s^2 - 9.8 m/s^2 = 5.2 m/s^2
How far will the rocket go?
The initial velocity of the rocket is zero. Given that, you learned in an earlier module
that the distance that the rocket will travel during the burn is
d = 0.5*Anet*t^2 = 0.5 * (5.2 m/s^2) * (10s)^2, or
d = 260 meters
260 meters straight up
In other words, when the rocket runs out of fuel at the end of the 10-second
burn, the rocket has traveled
straight up by 260 meters. Given that it started 15 meters above the ground, it
is at a height of 275 meters above the ground at that point in time.
Total mechanical energy
At that point in time, the total mechanical energy possessed by the rocket consists of
its gravitational potential energy plus its kinetic energy.
The kinetic energy
To compute the kinetic energy, we need to know the velocity. You learned in
an earlier module that we can compute
the velocity as
v = Anet * t = (5.2 m/s^2) * 10s = 52 m/s
You also learned earlier that the kinetic energy is equal to
KE = 0.5 * m * v^2 = 0.5 * 10 kg * (52 m/s)^2 = 13520 joules
Gravitational potential energy
You learned in an earlier module that the gravitational potential energy of
an object due to its height above the surface of the earth is equal to
PEg = m * g * h = 10 kg * (9.8m/s^2) * 275 m = 26950 joules
Note that this value is computed using the height above the ground and not
the height above the platform, which is 260 meters.
The total mechanical energy
Thus, the total mechanical energy at this point in time is
ME
= 13520 joules + 26950 joules = 40470 joules
Validation
Let's see if we can validate that result in some other way.
We know that the mechanical energy of the rocket at rest on the platform was
equal to 1470 joules.
We can compute the work done in moving the rocket up by 260 meters by
multiplying that distance by the upward force. Thus,
work = distance * thrust, or
work = 260 m * 150N = 39000 joules
Additional mechanical energy
This is the mechanical energy added to the rocket after it left the platform
while the engine was burning.
The total mechanical energy when the burn ends is the sum of that value and the
mechanical energy that it had while at rest on the platform. Thus, the total
mechanical energy at the end of the burn is:
ME = 1470 joules + 39000 joules = 40470 joules
A good match
This value matches the value computed
earlier
on the
basis of the height of the rocket above the surface of the earth and the
velocity of the rocket. Thus, the two approaches agree with one another up to
this point.
State at the end of Leg B
Therefore, at the end of Leg B,
The total mechanical energy
possessed by the rocket is equal to 40470
joules.
The gravitational potential energy is 26950 joules
The kinetic energy is 13520 joules
The rocket is out of fuel and is coasting upward with a velocity of 52
m/s.
The only force acting on the rocket is an internal downward force due to gravity,
which is equal to 10kg * 9.8m/s^2 = 98 newtons.
<emphasis id="Leg_C" effect="bold">
Leg C
</emphasis>
This is the part of the trip where the rocket coasts from its height at the
end of Leg B to the apex of its trip. The continued upward motion is due solely to its kinetic energy at the end
of Leg B.
From the end of Leg B when the rocket engine stops burning, until the rocket crashes on the surface of the earth, the only
forces acting on the rocket will be the internal force of gravity.
Total mechanical energy is conserved
Since
internal forces cannot change the mechanical energy possessed by an object, the
total mechanical energy for the rocket must remain at
40470 joules
for the remainder of the trip.
How long to reach the apex?
We can compute the time required for the rocket to reach the apex as
t = v/g = (52m/s)/(9.8m/s^2) = 5.31 seconds
How far will the rocket travel?
Knowing the time required to reach the apex, we can compute the distance to
the apex (during this leg only) as
d = v0*t - 0.5*g*t^2, or
d = (52m/s) * (5.31s) - (0.5) * (9.8m/s^2)*(5.31s)^2, or
d = 138 meters
An additional 138 meters
In other words, the rocket travels an additional 138 meters straight up after
the rocket-engine stops burning. This additional travel is due solely to the
kinetic energy possessed by the rocket at the end of the burn.
The total height of the apex
Adding 138 more meters to the height at the end of the burn causes the height at the apex to be
height at apex = 275m + 138m = 413 meters
Mechanical energy equals potential energy alone
At that point, the total mechanical energy is equal to the gravitational
potential energy because the rocket isn't moving and the kinetic energy has gone to zero.
Validation
We can compute the total mechanical energy at this point as
PEg = m*g*h = 10kg * (9.8m/s^2) * 413m = 40474 joules
This result is close enough to
the total mechanical energy
at the end of Leg
B to validate the computations. In this case, we determined the height using
time, velocity, and acceleration, and validated that height using work/energy
concepts.
State at the end of Leg C
At the completion of Leg C:
The rocket is at the apex at a height of 413 meters.
The total mechanical energy is 40470 joules.
The kinetic energy is 0 because for an instant, the rocket isn't moving.
The mechanical energy consists totally of gravitational potential
energy.
<emphasis id="Leg_D" effect="bold">
Leg D
</emphasis>
Leg D of the trip is fairly simple. The rocket falls for a
distance of 413 meters under the influence of the internal gravitational force.
No change in mechanical energy
Once again, because the force is an internal force, the total mechanical
energy cannot be changed by the work done by the force. However, the mechanical
energy can be transformed from potential energy to kinetic energy.
At the instant before the rocket strikes the ground, it must still have a
total mechanical energy value of 40470 joules.
Kinetic energy: 40470, potential energy: 0
At the instant before the rocket strikes the ground, all of the mechanical
energy has been transformed into kinetic energy. We can use that knowledge to
compute the velocity of the rocket right before it strikes the ground.
KE = 0.5*m*v^2, or
v^2 = KE/(0.5*m), or
v = (KE/(0.5*m))^(1/2) = (40470 joules/(0.5*10kg))^(1/2), or
terminal velocity
= v = 90 meters/sec
Thus, the terminal velocity of the rocket when it strikes the ground is 90
meters/sec straight down.
Validation
Let's see if we can validate that result using a different approach. Given the height
of the apex and the
acceleration of gravity, we can computer the transit time as
413m = 0.5*g*t^2, or
t^2 = 413m/(0.5*g), or
t = (413m/(0.5*g))^(1/2) = (413m/(0.5*9.8m/s^2))^(1/2), or
t = 9.18 seconds
Compute the terminal velocity
Knowing the time to make the trip to the ground along with the acceleration, we can compute the
terminal velocity as
v = g * t = (9.8m/s^2)*9.18s = 90 m/s
which matches the
terminal velocity
arrived
at on the basis of work and energy.
State at the end of Leg D
Therefore, at the end of Leg D, the rocket crashes into the ground. However,
an instant before the crash,
The total mechanical energy is 40470 joules.
The gravitational potential energy is 0.
The kinetic energy is 40470 joules.
The velocity is 90 m/s straight down toward the center of the earth.
<emphasis id="Run_the_program" effect="bold">
Do the calculations
</emphasis>
I encourage you to repeat the calculations that I have presented in this lesson to confirm that you get the same results. Experiment with the scenarios, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.
<emphasis id="Resources" effect="bold">
Resources
</emphasis>
I will publish a module containing consolidated links to resources on my
Connexions web page and will update and add to the list as additional modules
in this collection are published.
<emphasis id="Miscellaneous" effect="bold">
Miscellaneous
</emphasis>
This section contains a variety of miscellaneous information.
Housekeeping material
Module name: Relationships Among Kinematics, Newton's Laws, Vectors, 2D Motion, 2D Forces, Momentum, Work, Energy, and Power for Blind Students
File: Phy1220.htm
Keywords:
physics
accessible
accessibility
blind
graph board
protractor
screen reader
refreshable Braille display
JavaScript
trigonometry
potential energy
work
gravitational potential energy
elastic potential energy
kinetic energy
mechanical energy
total mechanical energy
power
watt
internal force
conservative force
external force
non-conservative force
Disclaimers:
Financial
: Although the Connexions
site makes it possible for you to download a PDF file for this
module at no charge, and also makes it possible for you to
purchase a pre-printed version of the PDF file, you should be
aware that some of the HTML elements in this module may not translate well into
PDF.
I also want you to know that I receive no financial compensation from the Connexions website even if you purchase
the PDF version of the module.
Affiliation
: I am a professor of Computer Information
Technology at Austin Community College in Austin, TX.
-end-