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Question: What would be the final velocity of the coupled railroad cars?

Answer: For this scenario, we have

v2 = (ma*va1 +mb*vb1)/(ma + mb), or

v2 = (ma*10)/2*ma = 5 m/s

The final velocity of the pair of coupled cars is half the initial velocity of the car that was moving.

Scenario #2: Now assume that due to loading, Car-A has twice the mass of Car-B, the initial velocity for Car-A is 10m/s and the initial velocity forCar-B is 0.

Question: What would be the final velocity of the coupled railroad cars?

Answer: For this scenario, we have

v2 = (ma*va1 +mb*vb1)/(ma + mb), or

v2 = (ma*va1)/(ma + 0.5ma), or

v2 = ma*10/1.5*ma = 6.67 m/s

When the car at rest is less massive than the car in motion, the final velocity is a little higher than when the two cars have the same mass.

Scenario #3: Finally, assume that due to loading, Car-A has twice the mass of Car-B, the initial velocity for Car-A is 10m/s and the initial velocity forCar-B is 5m/s.

Question: What would be the final velocity of each railroad car?

Answer: For this scenario, we have

v2 = (ma*va1 +mb*vb1)/(ma + mb), or

v2 = (ma*10 +0.5ma*5)/(1.5*ma), or

v2 = (10/1.5) + (0.5*5/1.5) = 8.33 m/s

When both cars are already moving in the same direction, the final velocity in that direction is greater than when one of the cars is stationary.

Center of mass examples

This section contains solutions to problems involving the center of mass.

Two objects

Two objects are located on a flat lawn with the following mass values and locations:

  • obj1m = 15 kg
  • obj1x = 1 m
  • obj1y = 5 m
  • obj2m = 3 kg
  • obj2x = 4 m
  • obj2y = 2 m

What are the coordinates of the center of mass?


A JavaScript script that will solve this problem is shown in Listing 2 .

Listing 2 . Center of mass for two objects.
<!---------------- File JavaScript02.html ---------------------><html><body><script language="JavaScript1.3">document.write("Start Script</br>"); //Create arrays for mass,xCoor, and yCoor values;var mass = new Array(15,3); var xCoor = new Array(1,4);var yCoor = new Array(5,2); //Declare a counter variable.var cnt = 0; //Use a loop to compute total mass by summing the individual// mass values. var massTotal = 0;for(cnt = 0;cnt<mass.length;cnt++){ massTotal += mass[cnt]; }//end for loop//Use a loop to compute x-coordinate of the center of mass // by summing the normalized sum of products.var cmX = 0; for(cnt = 0;cnt<mass.length;cnt++){ cmX += mass[cnt]*xCoor[cnt]/massTotal}//end for loop //Use a loop to compute y-coordinate of the center of mass// by summing the normalized sum of products. var cmY = 0;for(cnt = 0;cnt<mass.length;cnt++){ cmY += mass[cnt]*yCoor[cnt]/massTotal}//end for loop //Display the resultsdocument.write("massTotal = " + massTotal + " kg</br>"); document.write("cmX = " + cmX + " meters</br>"); document.write("cmY = " + cmY + " meters</br>"); document.write("End Script");</script></body></html>

The comments describe how the problem is solved.

The output is shown in Figure 4 .

Figure 4 . Center of mass for two objects.
Start Script massTotal = 18 kgcmX = 1.5 meters cmY = 4.5 metersEnd Script

The x-coordinate for the center of mass is 1.5 meters, and the y-coordinate for the center of mass is 4.5 meters.

Three objects

Three objects are located on a flat lawn with the following mass values and locations:

  • obj1m = 5 kg
  • obj1x = 1 m
  • obj1y = 1 m
  • obj2m = 5 kg
  • obj2x = 3 m
  • obj2y = 1 m
  • obj3m = 10 kg
  • obj3x = 2 m
  • obj3y = 3 m

What are the coordinates of the center of mass?

Use the code from Listing 2 but make the change shown in Listing 3 in order to populate the world withthree objects having different mass values and different coordinates.

Listing 3 . Change to add a third object.
//Create arrays for mass,xCoor, and yCoor values; var mass = new Array(5,5,10);var xCoor = new Array(1,3,2); var yCoor = new Array(1,1,3);

The output is shown in Figure 5 .

Figure 5 . Center of mass for three objects.
Start Script massTotal = 20 kgcmX = 2 meters cmY = 2 metersEnd Script

Do the calculations

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.


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 modulesin this collection are published.


This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: Force and Motion -- Momentum, Impulse, and Conservation of Momentum for Blind Students
  • File: Phy1160.htm
  • Revised: 10/02/15
  • Keywords:
    • physics
    • accessible
    • accessibility
    • blind
    • graph board
    • protractor
    • screen reader
    • refreshable Braille display
    • JavaScript
    • trigonometry
    • conservation of momentum
    • momentum
    • Newton's cradle
    • impulse
    • action and reaction

Financial : Although the openstax CNX site makes it possible for you to download a PDF file for the collection that contains thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.

You also need to know that Prof. Baldwin receives no financial compensation from openstax CNX even if you purchase the PDF version of the collection.

In the past, unknown individuals have copied Prof. Baldwin's modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing Prof. Baldwin as the author.Prof. Baldwin neither receives compensation for those sales nor does he know who doesreceive compensation. If you purchase such a book, please be aware that it is a copy of a collection that is freelyavailable on openstax CNX and that it was made and published without the prior knowledge of Prof. Baldwin.

Affiliation : Prof. Baldwin is a professor of Computer Information Technology at Austin Community College in Austin, TX.


Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Accessible physics concepts for blind students. OpenStax CNX. Oct 02, 2015 Download for free at https://legacy.cnx.org/content/col11294/1.36
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