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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

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
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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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