Collisions: Crash Course Physics #10

Summary

This script delves into the physics of collisions, exploring momentum and impulse as key factors in how objects interact. It clarifies the difference between elastic and inelastic collisions, emphasizing that momentum is always conserved, regardless of the collision type. The concept of center of mass is introduced as crucial for understanding motion, especially in objects with uneven mass distribution. The episode is educational, providing a foundational understanding of these principles with relatable examples.

Takeaways

• 🔍 Physics explores various phenomena including collisions, which are significant in understanding how objects interact.
• 📚 Two critical concepts in collision physics are momentum, defined as mass times velocity, and impulse, which is the integral of net force over time.
• 🎱 Collisions can be classified as elastic, where kinetic energy is conserved, or inelastic, where kinetic energy is not conserved but momentum is.
• 🤔 In real-world scenarios, perfect elastic collisions are rare due to energy losses typically as heat or sound.
• 🏓 The conservation of momentum is a universal principle in collisions, ensuring that the total momentum before and after a collision remains constant.
• 🔄 Impulse is a useful measure in collision analysis, as it helps to describe the change in momentum during the brief time of impact.
• 🧲 Perfectly inelastic collisions occur when objects stick together, resulting in maximum kinetic energy loss but still conserving momentum.
• 🌐 Center of mass is a crucial concept for understanding the motion of objects, especially those with uneven mass distribution.
• 📏 The center of mass can be calculated using the formula that sums the product of each mass and its distance from a reference point, divided by the total mass.
• 🔄 Newton's laws, particularly the second and third laws, play a foundational role in understanding the dynamics of collisions.

Q & A

What are the two main qualities that need to be considered when studying collisions in physics?

The two main qualities to consider when studying collisions in physics are momentum and impulse.

What is the relationship between Newton's second law and the concept of momentum?

Newton's second law states that the net force on an object is equal to its mass times its acceleration. Momentum is an object's mass times its velocity, and the net force is equal to the change in mass-times-velocity over time, which is the derivative of momentum with respect to time.

How is momentum defined in the context of physics?

Momentum in physics is defined as an object's mass times its velocity, representing the object's tendency to remain in motion.

What is impulse in physics, and how is it related to force?

Impulse in physics is the integral of the net force on an object over time, which represents the change in momentum. It is related to force as it is the product of the force applied and the time over which it acts.

How does the concept of impulse help in describing collisions?

Impulse helps in describing collisions because it accounts for the very quick changes in forces that occur during a collision, allowing us to calculate the overall effect of the collision on an object's momentum.

What is the difference between elastic and inelastic collisions?

Elastic collisions are those in which kinetic energy is neither created nor destroyed, while in inelastic collisions, kinetic energy is not conserved and is typically converted into other forms of energy like heat or sound.

Why is it said that elastic collisions are not found in real life?

Elastic collisions are not found in real life because there is always some energy loss in a collision, usually as heat or sound, which means that kinetic energy is not perfectly conserved.

What is the law that ensures the conservation of momentum in all collisions?

The conservation of momentum in all collisions is ensured by Newton's third law, which states that every action has an equal and opposite reaction.

What is a perfectly inelastic collision, and how does it differ from other inelastic collisions?

A perfectly inelastic collision is one where objects stick together upon impact, losing as much kinetic energy as possible to other forms of energy. It differs from other inelastic collisions in that it results in the objects becoming a single unit after the collision.

What is the center of mass, and why is it important in understanding how objects move?

The center of mass is the average position of all the mass in a system, and it is important in understanding how objects move because it helps to describe the motion of objects that do not have a uniform mass distribution, especially during collisions or rotations.

How can the position of the center of mass be calculated for a system with multiple masses?

The position of the center of mass for a system with multiple masses can be calculated using the equation that sums the product of each individual mass and its distance from a chosen starting point, divided by the total mass in the system.

Outlines

00:00

💥 Understanding Collisions and Momentum

This paragraph introduces the topic of collisions in physics, emphasizing that they are a significant aspect of the study, ranging from billiard balls to video game mishaps. It explains the importance of momentum and impulse in analyzing collisions, introduces the concept of center of mass, and references Newton's second law of motion. Momentum is defined as the product of an object's mass and velocity, illustrating its role in the collision process. Impulse is introduced as the integral of net force over time, essentially the change in momentum. The paragraph also distinguishes between elastic and inelastic collisions, noting that while kinetic energy may not be conserved in inelastic collisions, momentum always is, thanks to Newton's third law of motion.

05:01

🔍 Deep Dive into Elastic, Inelastic, and Perfectly Inelastic Collisions

The second paragraph delves deeper into the types of collisions, focusing on perfectly inelastic collisions where objects stick together upon impact, leading to the maximum loss of kinetic energy while still conserving momentum. It provides a practical example involving magnets to illustrate this concept. The paragraph also touches on the importance of the center of mass in understanding the motion of objects, especially when they are not evenly distributed or are complex in shape. The center of mass is described as the average position of all the mass in a system, and its calculation is essential for predicting the behavior of objects during collisions or when in motion. The summary concludes with a brief mention of the production credits for the video series.

Keywords

💡Momentum

Momentum is defined as an object's mass times its velocity, representing the object's quantity of motion. It is a key concept in the video, illustrating the tendency of objects to maintain their motion and is crucial in understanding collisions. For example, the script contrasts a fast-moving bag of leaves with a massive boulder to demonstrate the difference in momentum due to their varying masses.

💡Impulse

Impulse, represented by the letter J, is the integral of the net force on an object over time, essentially the change in momentum. It is highlighted in the video as a useful measure in collision scenarios where forces change rapidly. The script uses the example of a ball hitting a wall to explain how impulse is calculated, emphasizing its importance in analyzing collisions.

💡Elastic Collision

An elastic collision is one where kinetic energy is conserved, meaning it is neither created nor destroyed. The video uses the example of two billiard balls colliding to illustrate this concept, where ideally, all kinetic energy from one ball is transferred to another, with no loss in the system.

💡Inelastic Collision

Inelastic collisions are those in which kinetic energy is not conserved and some of it is transformed into other forms of energy like heat or sound. The video clarifies that while momentum is still conserved in these collisions, there is a loss of kinetic energy, contrasting it with elastic collisions.

💡Perfectly Inelastic Collision

A perfectly inelastic collision is a specific type of inelastic collision where objects stick together upon impact, losing as much kinetic energy as possible. The video provides the example of magnets colliding and sticking together, resulting in the conservation of momentum but a significant loss of kinetic energy.

💡Center of Mass

The center of mass is the average position of all the mass in a system and is vital for understanding how objects move, especially when they are not evenly distributed. The video explains it with examples of a stick with balls of different masses on either end, illustrating how the center of mass shifts based on the distribution of mass.

💡Newton's Second Law

Newton's second law, as discussed in the video, states that the net force on an object is equal to its mass times its acceleration. However, the video clarifies that Newton originally described it in terms of 'quantity of motion,' which is mass times velocity, and how this relates to momentum.

💡Newton's Third Law

Newton's third law, mentioned in the video, states that every action has an equal and opposite reaction. This principle is applied to collisions, explaining how the forces exerted by colliding objects on each other are equal and opposite, which ties into the conservation of momentum.

💡Conservation of Momentum

The conservation of momentum principle states that in any closed system, the total momentum before and after a collision remains constant. The video uses this principle to explain how to calculate the velocities of objects after a collision, emphasizing its fundamental role in collision physics.

💡Kinetic Energy

Kinetic energy is the energy of motion, and it is a focal point in the video when discussing collisions. The script explains how kinetic energy behaves in elastic and inelastic collisions, noting that it is conserved in the former but not in the latter, where it is transformed into other forms of energy.

💡Derivative

The concept of a derivative, while not explicitly named in the script, is implied when discussing Newton's second law in terms of the rate of change of mass-times-velocity with respect to time. It is a mathematical operation that relates to the calculation of momentum and is fundamental in physics for understanding how quantities change over time.

Highlights

Physics explores not only inclined blocks and floating astronauts but also collisions.

Momentum and impulse are key concepts in understanding collisions.

Momentum is an object's mass times its velocity and represents its resistance to stop.

Impulse is the integral of net force over time and represents a change in momentum.

Collisions can be elastic, where kinetic energy is conserved, or inelastic, where kinetic energy is lost.

Elastic collisions are idealized and rarely occur in real life due to energy loss as heat or sound.

Inelastic collisions result in a loss of kinetic energy, but momentum is always conserved.

Perfectly inelastic collisions occur when objects stick together, maximizing energy loss.

Newton's laws of motion, particularly the second and third, are fundamental to collision dynamics.

Newton's second law can be expressed in terms of momentum, highlighting its importance in motion.

The center of mass is a crucial concept for understanding the motion of objects with uneven mass distribution.

The center of mass is the average position of mass in a system and influences how objects move and rotate.

Calculating the center of mass involves summing the product of individual masses and their distances from a reference point.

Objects like hammers and pendulums with uneven mass distribution behave differently based on their center of mass.

Understanding momentum, impulse, and the center of mass is essential for analyzing collisions and object motion.

Crash Course Physics is produced in collaboration with PBS Digital Studios, offering a variety of educational content.

Transcripts

You and I have explored the rules that govern lots of different moving objects so far.

But physics isn’t just about dragging blocks up inclines, or astronauts floating in Vomit Comets.

There are also ... collisions!

And physics has a lot to say about collisions -- whether it’s two billiard balls knocking

against each other, or what happens when you fail at a Super Mario level for the 47th time

and throw your controller at the floor.

Stupid lava sticks!

To figure out what’s happening when objects collide, we’ll have to take into account

two main qualities: momentum and impulse.

We’ll also discuss what physicists mean when they talk about center of mass, and why that’s important.

And we’ll have our old friend Isaac Newton to help us out along the way.

[Theme Music]

Remember Newton’s second law?

That’s the one that says the net force on an object is equal to its mass, times its acceleration.

Except, that’s not actually what Newton said.

He really said that an object’s so-called “quantity of motion” was equal to its mass, times its velocity.

And the net force is equal to the change in that mass-times-velocity over time.

In other words, it’s the derivative of mass-times-velocity with respect to time.

And if you were to calculate that derivative, you’d find that the net force is just equal to mass times acceleration.

But putting Newton’s second law in terms of mass and velocity introduces an aspect

of motion that we haven’t talked about yet.

Newton didn’t really give this aspect a name, but we will: it’s called momentum,

and it’s one of those things that’s easier to see in real life than to describe.

Momentum is often described as an object’s tendency to remain in motion, but technically,

it’s an object’s mass times its velocity.

So, a big bag full of leaves rolling down a hill? It might be going fast, but it doesn’t

have much mass, so it doesn’t have a lot of momentum, and it wouldn’t be too hard to stop.

But the boulder chasing Indiana Jones? That had a lot of mass -- and therefore lots of momentum.

So it would have been much harder to stop.

And momentum is one factor that affects collisions between objects.

After all, if a huge boulder crashes into another huge boulder, that’s going to be

a very different sort of crash than if a bag of leaves crashes into a boulder.

But the other quality of a collision that we often consider is known as impulse, which

-- at least in the context of physics -- doesn’t actually have anything to do with willpower,

or why you throw your game controller when you get stuck on a level.

Instead, impulse -- usually represented by a J -- is the integral of the net force on

an object over time -- in other words, its change in momentum.

Impulse turns out to be a particularly useful way to describe a crash --

because generally, in collisions, forces change very quickly.

So, if a ball smacks into a wall, and over the course of half a second, its force on

the wall in Newtons is equal to the time, multiplied by 25, we’d say that its impulse was 3.1 Newton-seconds.

Now, let’s consider the different kinds of collisions that we can study.

Generally, collisions can be described as either elastic or inelastic.

And it’s going to be important to figure out which kind you’re dealing with, because the math works in very different ways.

If elastic collisions sound bouncy, that’s because they are.

Like the conservative systems we talked about last time, in elastic collisions, kinetic energy is neither created nor destroyed.

For example: let’s say you knock a white billiard ball into a second, red one that’s

sitting on the table, and they hit each other in just the right way.

For this to be a true elastic collision, all of the kinetic energy from the white ball

would be transferred to the red ball.

Meaning, after they hit each other, the white ball would stay put, and the red one would

zoom away with all of the kinetic energy -- so, the same speed, basically -- that the white ball used to have.

But you won’t come across elastic collisions in real life.

Because: There’s always going to be some energy that’s lost somewhere in a collision,

generally as heat or sound.

And when kinetic energy isn’t conserved, that’s an inelastic collision.

There’s one thing that’s going to be true about every collision, though, whether or not it’s elastic:

The momentum of the system will always be conserved.

It might be transferred to another object -- it might even be transferred to more than one object --

but the momentum is always going to go somewhere,

And we’ll be able to use math to figure out where it went.

And we can use what we know about impulse -- and Newton’s third law -- to prove it.

The third law, of course, is the one that says that every action has an equal and opposite reaction.

And that applies to collisions in the sense that, if a ball hits a wall, it’ll exert a force

on the wall, and the wall will exert an equally strong force on the ball.

We can describe each of these forces as impulses, since we know that an impulse is just

a change in an object’s momentum.

So, the ball’s momentum will be decreasing when it hits the wall. But because of Newton’s

third law, we know that the wall’s momentum is going to increase by an equal amount.

The change in the wall’s momentum might be impossible for us to see, because the wall

is connected to the ground, and Earth has lots of mass. But it’s there.

And that fact -- that momentum is always conserved -- turns out to be super helpful for describing collisions using math.

Like in the case where you knocked the white billiard ball into the red one.

Since momentum is conserved, and momentum is mass times velocity, the white ball’s

mass-times-velocity before the collision has to be equal to its mass-times-velocity, plus

the red ball’s mass-times-velocity, after the collision.

Which is why -- assuming the balls have the same mass -- if the white ball stops moving

after the collision, then the red ball must move with the same velocity that the white ball had.

So, now we know about both elastic and inelastic collisions.

But there’s also such a thing as a perfectly inelastic collision, because... of course there is.

And it’s easier for me to tell you, first, what it isn’t.

So, it is not a collision where the objects lose all of their kinetic energy.

Instead, a perfectly inelastic collision is what happens when objects stick together.

These collisions lose as much kinetic energy as possible to other forms of energy, like

heat, sound, or even potential energy ... but still, their momentum is conserved.

An example would be if you pushed one magnet toward another -- at just the right angle

for them to stick together on contact -- and then they both started sliding together at

half the speed of the magnet you pushed.

Before the collision, the momentum of one magnet was zero, and the momentum of the one

you pushed was its mass times its velocity.

Once the magnets collide, the mass is doubled, and the velocity is cut in half.

So the total momentum stays the same, but you lose some kinetic energy because there’s less speed involved.

So that’s the basics of how collisions work, and how they relate to the momentum of motion in a straight line.

But there’s one more detail we have to explore in order to really understand how objects move,

whether they’re going to collide or not.

And that is: center of mass.

Until now, we’ve been talking about objects as though they were little point-particles.

And that’s worked fine -- for the most part, the objects we’ve been talking about would act much like a small dot would.

But of course, not all objects work that way.

If you’ve ever tried to fling a hammer, for example -- which I do not recommend doing!

-- it wouldn’t fly through the air the same way a softball would, because the hammer’s mass isn’t distributed evenly.

Likewise, a pendulum with a big ball on the end of a very light string -- called simple pendulum --

would behave very differently from a pendulum that uses a heavier stick -- what’s known as a physical pendulum.

In these situations, it’s more useful to describe what the center of mass is doing.

When you throw the hammer, for example, it’ll rotate around its center of mass.

So, what is the center of mass?

It’s basically the average position of all the mass in the system.

Say you have a 3 meter long stick -- which we’ll pretend is massless -- with a 2-kilogram ball stuck on either end.

It’s easy to see where the center of mass should be -- the mass is distributed symmetrically,

so its center is going to be right in the middle of the stick.

Now, let’s say you have another 3 meter stick, and on the left side there’s a 2

kilogram ball, but on the right side, there’s a 4 kilogram ball.

This time, there’s twice the mass on the right side of the stick.

So when you’re trying to calculate the average position of all the mass attached to the stick,

you’re going to be counting the right-hand side twice as much.

That means the center of mass will be two thirds of the way along the stick,

closer to the 4 kilogram ball.

It’s like each piece of mass pulls on the center of mass a little bit, so parts with

more mass end up pulling harder and moving it closer.

But if you don’t want to calculate this in your head -- and if there are like seven

different particles to deal with, you probably won’t -- but there’s an equation for it!

First, pick a starting point to measure from, where x = 0.

That can be the end of the stick, the middle of the stick, whatever’s easiest.

As long as you’re consistent.

Then, the center of mass will be equal to the sum of each individual mass, times its

distance from the starting point, all divided by the total mass in the system.

Let’s try it for our stick with the differently-weighted balls.

We’ll choose the left side of the stick, where the 2 kilogram ball is, as our starting point.

The 2 kilogram ball’s mass times its position is zero.

The 4 kilogram ball’s mass times its position -- 3 meters -- is 12 kilogram-meters.

And the total mass of the system is 6 kilograms.

So, divide 12 kilogram-meters by 6 kilograms, and you get 2 meters.

That’s the position of the center of mass, which is two thirds of the way along the stick,

toward the 4 kilogram ball’s side. Exactly what we figured out earlier.

I’m telling you all of this now, because from here, we’re heading off in a totally new direction.

Literally!

But for now, you learned about collisions, and how momentum and impulse can be used to describe them.

We also talked about the differences between elastic and inelastic collisions,

and how to calculate the center of mass of a system.

Crash Course Physics is produced in association with PBS Digital Studios. You can head over

to their channel to check out amazing shows like Physics Girl, Gross Science and The Chatterbox.

This episode of Crash Course was filmed in the Doctor Cheryl C. Kinney Crash Course Studio

with the help of these amazing people and our equally amazing graphics team is Thought Cafe.