Author Archives: Todd Helmenstine

About Todd Helmenstine

Todd Helmenstine is the physicist/mathematician who creates most of the images and PDF files found on sciencenotes.org. Nearly all of the graphics are created in Adobe Illustrator, Fireworks and Photoshop. Todd also writes many of the example problems and general news articles found on the site.

Elastic Collision of Two Masses – It Can Be Shown Exercise

An elastic collision is a collision where total momentum and total kinetic energy is conserved.

Elastic Collision - Conservation of Momentum Example

This illustration shows two objects A and B traveling towards each other. The mass of A is mA and the moving with velocity VAi. The second object has a mass of mB and velocity VBi. The two objects collide elastically. Mass A moves away at a velocity VAf and mass B has a final velocity of VBf.

Given these conditions, textbooks give the following formulas for VAf and VBf.

Elastic Collision Final Velocity of Mass A Formula
and
Elastic Collision Final Velocity of Mass B Formula

where
mA is the mass of the first object
VAi is the initial velocity of the first object
VAf is the final velocity of the first object
mB is the mass of the second object
VBi is the initial velocity of the second object and
VBf is the final velocity of the second object.

These two equations are often just presented in this form in the textbook with little or no explanations. Very early in your science education, you will encounter the phrase “It can be shown …” between two steps of mathematics or “left as an exercise for the student”. This almost always translates into “homework problem”. This “It can be shown” example shows how to find the final velocities of two masses after an elastic collision.

This is a step by step derivation of these two equations.

First, we know total momentum is conserved in the collision.

total momentum before collision = total momentum after collision

mAVAi + mBVBi = mAVAf + mBVBf

Rearrange this equation so the same masses are on the same side as each other

mAVAi – mAVAf = mBVBf – mBVBi

Factor out the masses

mA(VAi – VAf) = mB(VBf – VBi)

Let’s call this Equation 1 and come back to it in a minute.

Since we were told the collision was elastic, the total kinetic energy is conserved.

kinetic energy before collision = kinetic energy after collection

½mAVAi2 + ½mBVBi2 = ½mAVAf2 + ½mBVBf2

Multiply the entire equation by 2 to get rid of the ½ factors.

mAVAi2 + mBVBi2 = mAVAf2 + mBVBf2

Rearrange the equation so the like masses are together.

mAVAi2 – mAVAf2 = mBVBf2 – mBVBi2

Factor out the common masses

mA(VAi2 – VAf2) = mB(VBf2 – VBi2)

Use the “difference between two squares” relationship (a2 – b2) = (a + b)(a – b) to factor out the squared velocities on each side.

mA(VAi + VAf)(VAi – VAf) = mB(VBf + VBi)(VBf – VBi)

Now we have two equations and two unknowns, VAf and VBf.

Divide this equation by equation 1 from before (the total momentum equation from above) to get

Elastic Collision Math Step 1

Now we can cancel out most of this

Elastic Collison Math Step 2

This leaves

VAi + VAf = VBf + VBi

Solve for VAf

VAf = VBf + VBi – VAi

Now we have one of our unknowns in terms of the other unknown variable. Plug this into the original total momentum equation

mAVAi + mBVBi = mAVAf + mBVBf

mAVAi + mBVBi = mA(VBf + VBi – VAi) + mBVBf

Now, solve this for the final unknown variable, VBf

mAVAi + mBVBi = mAVBf + mAVBi – mAVAi + mBVBf

subtract mAVBi from both sides and add mAVAi to both sides

mAVAi + mBVBi – mAVBi + mAVAi = mAVBf + mBVBf

2mAVAi + mBVBi – mAVBi = mAVBf + mBVBf

factor out the masses

2 mAVAi + (mB – mA)VBi = (mA + mB)VBf

Divide both sides by (mA + mB)

elastic collision math step 3

Elastic Collision math final form of final velocity of second mass

Now we know the value of one of the unknowns, VBf. Use this to find the other unknown variable, VAf. Earlier, we found

VAf = VBf + VBi – VAi

Plug in our VBf equation and solve for VAf

Elastic Collision Step 1 solve for final velocity of object A

Group the terms with the same velocities

Elastic Collision step 2 solving for final velocity of mass A

The common denominator for both sides is (mA + mB)

elastic collision step 3 solving for final velocity of mass A

elastic collision step 4 solving for final velocity of mass A

Be careful of your signs in the first half of the expressions in this step

elastic collision step 5 solving for final velocity of mass A

Elastic Collision Final Velocity of Mass A Formula

Now we’ve solved for both unknowns VAf and VBf in terms of known values.

Elastic Collision Final Velocity of Mass A Formula

Elastic Collision Final Velocity of Mass B Formula

Note these match the equations we were supposed to find.

This was not a difficult problem, but there were a couple spots to trip you up.

First, all the subscripts can get tangled up if you aren’t careful or neat in your handwriting.

Second, sign errors. Subtracting a pair of variables inside parentheses will change the sign on BOTH variables. It is all too easy to carelessly turn – (a + b) into -a + b instead of -a – b.

Last, learn the difference between two squares factor. a2 – b2 = (a + b)(a – b) is an extremely useful factoring trick when trying to cancel something out of an equation.

Impulse and Momentum – Physics Example Problem

Desktop Momentum Balls Toy

This model of a common desktop toy shows the forces acting on the raised ball. The principles of impulse and momentum show how the momentum is transferred to each ball and the process repeats.

Impulse and momentum are physical concepts that are easily seen from Newton’s Laws of Motion.

Start with this equation of motion for constant acceleration.

v = v0 + at

where
v = velocity
v0 = initial velocity
a = acceleration
t = time

If you rearrange the equation, you get

v – v0 = at

Newton’s second law deals with with force.

F = ma

where
F = force
m = mass
a = acceleration

solve this for a and get

a = F/m

Stick this into the velocity equation and get

v – v0 = (F/m)t

Multiply both sides by m

mv – mv0 = Ft

The left side of the equation deals with momentum (often denoted by a lower-case p) and the right side is impulse (often denoted by an upper-case letter J).

Mass times velocity is known as momentum and force applied over time is called impulse.

Impulse and Momentum Example Problem

Question: A 50 kg mass is sitting on a frictionless surface. An unknown constant force pushes the mass for 2 seconds until the mass reaches a velocity of 3 m/s.

a) What is the initial momentum of the mass?
b) What is the final momentum of the mass?
c) What was the force acting on the mass?
d) What was the impulse acting on the mass?

Impulse and Momentum Example Problem

Part a) What is the initial momentum?

Momentum is mass times velocity. Since the mass is at rest, the initial velocity is 0 m/s.

momentum = m⋅v = (50 kg)⋅(0 m/s) = 0 kg⋅m/s

Part b) What is the final momentum?

After the force is finished acting on the mass, the velocity is 3 m/s.

momentum = m⋅v = (50 kg)⋅(3 m/s) = 150 kg⋅m/s

Part c) What was the force acting on the mass?

mv – mv0 = Ft

From parts a and b, we know mv0 = 0 kg⋅m/s and mv = 150 kg⋅m/s.

150 kg⋅m/s – 0 kg⋅m/s = Ft
150 kg⋅m/s = Ft

Since the force was in effect over 3 seconds, t = 3 s.

150 kg⋅m/s = F ⋅ 3s
F = (150 kg⋅m/s) / 3 s
F = 50 kg⋅m/s2

Unit Fact: kg⋅m/s2 can be denoted by the derived SI unit Newton (symbol N)

F = 50 N

Part d) What was the impulse acting on the mass?

The impulse is the force multiplied by the time passed. It is also equal to the change in momentum over the same time period.

Ft = 50 N ⋅ 3 s
Ft = 150 Ns or 150 kg⋅m/s

The impulse was 150 kg⋅m/s.

These problems are relatively simple as long as you keep your units straight. Impulse and momentum should have the same units: mass⋅velocity or force⋅time. Check your units when you check your answer.

Another possible way to cause errors is to confuse your vector directions. Velocity and Force are both vector quantities. In this example, the mass was pushed in the direction of the final velocity. If another force pushed in the opposite direction to slow down the mass, the force would have a negative value compared to the velocity vector.

If you found this helpful, check out other Physics Example Problems.

Neon Rainbow Periodic Table Wallpaper

This rainbow periodic table wallpaper is a colorful addition to our collection of wallpapers.

Neon Rainbow Periodic Table WallpaperThe periodic table contains each element’s number, symbol, name, and atomic mass. It also includes the four new elements: nihonium (113), moscovium (115), tennessine (117), and oganesson (118).

Thiswallpaper is 5760 x 3240 pixels and can be scaled down to look great on any HD device. Click the image to view the full-sized image, or download a copy directly.

We’ve partnered with Redbubble to make posters of many of our periodic tables. This one is no exception. Check it out!  They’ve even got t-shirts, pillows, and coffee cups,

Periodic Table For Kids With 118 Elements

These two “periodic table for kids” tables are simplified tables containing each element’s atomic number, symbol, name and atomic mass rounded to two decimal places. These two tables include the four new elements: nihonium (113), moscovium (115), tennessine (117), and oganesson (118).

Color Periodic Table for Kids

Color Periodic Table For Kids - 2017 EditionThis first table is a color version where the colors correspond to the different element groups.

For easier printing, a PDF is available. The table is optimized to fit on a single 8½ x 11″ sheet of paper.

Black and White Periodic Table for Kids

Printable Periodic Table For Kids - 2017 EditionThis table is better for those without access to color printers or wishes to leave coloring to the student.

As with the color version, a PDF is available for downloading and printing.

If you enjoy these periodic tables, check out our many other periodic tables.

Periodic Table In Black and White Wallpaper

This wallpaper periodic table in black and white is a simple and clean design.

Periodic table in black and whiteEach element is represented by its atomic number, symbol, name, and atomic mass. This periodic table also includes the four new Period 7 elements: nihonium, moscovium, tennessine, and oganesson.

This wallpaper is optimized to look great on a 4K display at 3840 x 2160 pixels but looks great on HD monitors and devices when scaled down. Click the image or download to view the full-sized image. If you’d like to print a copy, download the PDF and set your options to “Fit” and “Landscape” for best results.

Periodic Table In White and BlackThis periodic table is the same table, just reversed. The text and borders are white and the background is black. Click the image or download directly.

Neon Periodic Table With 118 Elements Wallpaper

If you look closely, this neon periodic table wallpaper shows each element surrounded by colored neon bulbs.

Neon Periodic Table - 2017 Edition (118 Elements)The different colors correspond to the different element groups. Match the group to the color using the legend along the bottom. Each element is represented by its number, symbol, name, and atomic mass. This table also includes the four new Period 7 elements: nihonium, moscovium, tennessine, and oganesson.

This wallpaper is optimized to look great on a 4K display at 3840 x 2160 pixels but looks great on HD monitors and devices when scaled down. Click the image or download to view the full-sized image. If you’d like to print a copy, download the PDF and set your options to “Fit” and “Landscape” for best results.

Some folks find a white background a little hard on the eyes for their monitors. This periodic table is the same table with a black background and high-contrast white text. Click the image or download directly.

Neon Periodic Table With Black Background - 2017 Edition

Large Print Periodic Table

No time for fine print. This large print periodic table will help you with your eye strain.

Color Large Print Periodic Table

Color Large Print Periodic TableThis color periodic table contains the usual element numbers, symbols, element names, and atomic masses, but has the added bonus of larger type. This periodic table includes all 118 elements known as of 2017.

The element names are as large as I could consistently make them. I chose to make the names a uniform font size with the exception of the really long ones (rutherfordium, darmstadtium, roentgenium, and praseodymium). The rest of the information is much easier to read than the usual printable periodic table.

To print, download the PDF and choose either “Fit” or “Shrink oversized pages” under Page Sizing and Handling in the print options. Remember to choose Landscape for your orientation to use more space on the paper.

Black and White Large Print Periodic Table

This is basically the same table, only monochromatic!

Large Print Periodic TableDownload this PDF if you don’t have access to a color printer, or just want to save your color inks or toners. Choose the same options as the color periodic table to print on a single sheet of paper.

If you like this table, check out our other printable periodic tables.