High School Physics Collisions & Linear Momentum

Hi, welcome back to educator.com, today we are going to be talking about collisions and linear momentum.0000

One really great importance of Physics is, being able to understand the interactions between the objects, the way things work between one another in the world.0006

One of the most basic forms of interaction is collision.0015

When two things hit together, it is useful to be able to know how things are going to play out at that moment.0018

If you can model it, we have done all sorts of useful things.0024

Think of all the things that we can do with it.0026

We can investigate a car crash, you could dock a shuttle with a space station, you could look at the collisions between atomic particles.0027

There is all sorts of stuff, stuff bounces off one another, where stuff hit one another and sticks together, you will be able to understand it a lot better if you have momentum and how it is conserved, how it works for momentum.0034

Investigating collisions!0046

If you wanted to talk more about collisions, what ideas would be useful?0048

We can all figure out at this point , momentum is going to be what we are going to be talking about.0052

We just introduced the idea of momentum, and the reason why we did that was to set ourselves up to be talking about conservation of momentum, and how it works with collisions.0058

If we are going to investigate collisions, if we are going to investigate how motion changes, and what is happening as things go on, we want to be able to talk about how motion is working.0066

One of the best things we have for talking about motion is momentum, p = mv.0076

If we got that under our belts, that is going to really help us.0080

But that is not quite enough, we need one more thing.0084

The other thing we need, is the centre of mass.0087

Talking about the centre of mass allows us to explore the motion of a system, how things work together, and that is what we are going to need to be able to talk about collision.0090

Two objects working together, hitting each other, that is the system, you look at those two objects and the way they interact together, they are going to have to interact as a system because we have decided to make them a system.0097

If you look at them as a system, we will be able to know things, the way their momenta are working, and the way their centre of mass is doing things.0106

We are going to use to use centre of mass to learn indirectly more about momentum.0113

We are going to derive this.0117

We can extend the idea of momentum to a system, by looking at the total of the momenta in the system, makes up the momentum of the system.0118

This symbol right here mean, ADD things together. (Might be a little confusing if you have not seen it before.)0126

That says, add up all the momenta in the system.0135

Say we are looking at a pretty basic system that has some p₁, and p₂, a two object system.0139

Then the momentum of that system is adding them together, and that makes sense, the final momentum of the system is just all of them added together.0146

Just like we said that the energy of the system is all of the energies involved, you look at the energy for, potential gravity, kinetic energy, spring energy, you added them all to find out what the energy of the system was.0153

Exact same thing here: To find out the momentum of a system, you just add all the momenta together.0167

How do we know for sure though, that the mass of the system, (all masses summed up) times the velocity of the centre of mass, is equal to the momentum of the system (all added together)?0173

We can actually prove that.0185

M_sysv_CM = (m₁+m₂)(m₁v₁+m₂v₂)/(m₁+m₂) (in 2 particle system, what happens when two objects interact with one another, this works with an arbitrary number of things in our system, but we are going to be looking at the collision between just two things, to understand how this came to be, also note that velocity of centre of mass is the weighted average.)0186

We have got something that cancels off pretty easily, and from there, (m₁v₁+m₂v₂) = p₁+p₂ = Σp.0255

So, all the momenta in the system added up together, is equal to the momentum of the system, which turns out to be, = M_sysv_CM.0290

We look at the thing as a whole, it is the mass of the system times the movement of the system of a whole, or we look at the pieces that make it up, it is all those pieces added together, great intuition, works out perfectly.0299

Similarly, we are not going to show this mathematically, but we can do this with impulse.0312

The change in momentum of the system, Δp_sys = j_sys = F_extt.0316

That makes sense, if you are going to have a change in the system, you are going to put an impulse on the system, just like we did when we were working with single object at a time, but it is also going to be the case that the force on it is external, because remember, form our work with centre of mass, internal forces, because of newton's third law, every action has an equal and opposite reactionary force,0322

So, if it is an internal force in the system, sure, stuff is going to happen inside of the system, but the ultimate change to the centre of mass, is going to get a force going one way, but it is going to get the exact same force going the other way, they will cancel out.0343

Centre of mass is going to experience nothing.0355

So, form our work in centre of mass, we know that internal forces have no effect on the system's centre of mass, so no effect on the system's momentum.0356

So, Δp_sys = j_sys = F_extt.0362

Now we are ready to make something really useful.0372

Conservation of linear momentum: If there are no external forces acting on the system, there is no net external force acting on the system, and there should not be external forces acting on the system, but as long as they are going to cancel out, at static equilibrium, we are going to have the change in momentum of the system = zero.0375

That means, system's momentum at the beginning = system's momentum at the end.0394

If we know that there is no net external force, we know that we are going to have a constant momentum, a conserved linear momentum.0401

That is really handy, because that is going to give us a whole bunch of power, when we are looking at new kinds of problems.0409

Once again, notice that there can be internal forces, it does not matter if there are internal forces, they will have no effect on the system's momentum as a whole.0416

For example, if we had a bomb, or some explosive things, and we blew it up into a bunch of pieces.0422

Each one of those pieces would fly away.0436

But, since it is all internal forces in that system, the system's momentum is going to wind up being that same steady state when it was at rest, it is going to be zero.0439

Of course, there is also gas in there, so it is pretty complicated thing to model, you might have to deal with the gas moving in every which way as well, but the total system's momentum, when we take into account the pieces flying every way, the gas in the centre expanding out rapidly, the end of it is going to wind up also being zero momentum.0450

All sorts of things are going in opposite directions, but when we add them all up together, they ultimately cancel each other, and we get back to rest.0470

Conservation and external force:0477

This means, if we have an external force acting on our system, we require to have no external force to allow the conservation of momentum, so fi there are external forces, we fling up our hands, and go, 'Oh, we cannot do this problem'?0479

No, we do not have to do that, because we can still examine a collision, even when there is an external force. Why?0491

External forces are normally pretty small compared to collision forces.0496

Think about, if a car was crashing into another car, there is going to be little bit of friction on the ground, but for the amount of time that the crash occurs, friction on the road, like small amount of rolling resistance, small amount of internal friction inside the engine, it is going to be very little compared to the massive change in forces involved in two cars crashing together.0501

Those massive forces are going to be so large compared to all the other forces in effect, that we can basically forget about those other forces, we can forget about those external forces, because we are going to be looking at a brief period of time.0527

We can take a snap shot just before the incident of impact, a snap shot just after the incident of impact, because the impact is going to be more than an instant, but it will be something very brief, order of (1/10) of a second or less.0542

We can take a look just before the impact happens, and just after the impact is complete, the external forces are likely to be pretty small compared to the forces involved in the collision, so we can still effectively use conservation of momentum.0555

It is not perfectly correct, but it is so close as to be plenty good.0570

We are going to be totally be able to use it, and it will still make a lot of sense.0576

As long as we consider the instant just before and just after the collision, we can treat it as the momentum is conserved during the collision.0578

It will not be perfectly conserved, but it will be such a small change, that it is as if it was conserved, so it is good enough for us.0584

Finally, momentum versus energy:0593

It is really important to notice that momentum and energy are totally different.0596

mv is very different from (1/2)mv², conservation of momentum does not imply conservation of the other.0599

In fact, in almost all every day collisions, some energy is lost to heat via friction.0607

When, for example, you drop a ball, the ball hits the ground and rebounds.0611

If it had the same exact amount of energy that it started with, the amount of energy in that ball would have to be the same as what it was when it was dropped in the beginning.0617

That means, that it would be able to bounce and come up to the same height, again and again, but it does not happen.0625

For anybody who has dropped a ball, if you drop a ball, it does not keep bouncing forever, it eventually stops moving.0632

If it stops moving, the energy is being sapped.0642

Momentum is conserved, when you consider the earth and the ball interacting with one another, but the energy is not conserved in the ball-earth system.0644

The ball-earth system winds up dissipating energy.0654

That energy does get dissipated, conservation of energy does still work, but it winds up being dissipated into things that we cannot really watch, like heat energy, sound energy, so we do not really have a good way of doing it.0658

So from our point of view, mechanical energy is not conserved, the way that we know how to deal with energy, just is not conserved.0669

So we cannot use it, we can only keep momentum in that case.0675

The system's momentum will remain constant through a collision, but the energy of the system has no such restrictions.0678

When we are dealing with collisions, we know for sure, that momentum is going to stay the same.0685

But, energy might change.0689

Similarly, when we throw an object, we know that the energy in the object is going to remain the same through its motion.0693

However, since there are external forces, like gravity acting on the object, we know that the momentum of the object is not going to be the same.0699

Sometimes energy will be conserved, with momentum not being conserved, sometimes momentum will be conserved with energy not being conserved.0706

It depends on what you are looking at, you want to think about it as how short is the time frame, what external forces are acting on it, think in terms of collisions, means momentum you can trust, and no external forces means that we can trust in energy.0712

And external forces and energy, remember, gravity is not an external force for energy, because we have taken care of it within our energy formula.0730

But for momentum, there is no such formulae for these other things, like gravity or friction.0737

Still, finally, there are some collisions where energy is not lost.0742

So, we are going to have to categorize our collisions into different types.0747

When momentum is kept, when energy is kept; when momentum is kept, when energy is not kept.0749

Fist one, elastic: Collision is called elastic when energy is conserved in addition to momentum.0754

The change in the energy of the system = zero, and the change in the momentum of the system = zero.0760

These are really uncommon in everyday activities.0764

Stuff that we are used to in reality, in real life, it pretty much never happens, if not completely never.0767

Inelastic: Collisions that we do see in everyday life.0774

A collision is called elastic when energy is not conserved during the collisions.0777

So, we cannot trust the energy of the system to be equal to the energy of the system at the end.0780

Beginning system energy, not equal to ending system energy.0784

But, the change in the momentum of the system, is trustable, we can keep that.0790

The beginning of the collision, the end of the collision, they are going to wind up having the same momentum on both sides.0794

Those are called inelastic collisions.0799

When energy is not conserved, inelastic.0802

Finally, in addition, when object involved sticks together after the collision, when they hit each other and they just keep moving as just one thing, it is called completely inelastic.0805

They just do not hit each other, and lose some energy, they hit each other, and get turned into one object, or something that can be treated as one single object.0814

Finally, almost all everyday collisions are inelastic, sometimes it is useful to approximate some collisions as elastic.0823

However, in the atomic world, particles often have elastic collisions.0830

In the atomic world, elastic collisions are actually really common and standard.0833

But, form the sort of thing that we see in everyday life, we are almost never going to see an elastic collision.0836

However there are certain things that we can approximate as elastic collisions, and learn some more stuff about.0841

On to the examples.0847

Two cars are being driven directly towards one another.0849

They are remote controlled, so nobody is hurt in the process of this example!0851

First car has mass, m₁ = 1000 kg, and velocity, v₁ = 30 m/s.0855

Second car has mass, m₁ = 2000 kg, and traveling in the opposite direction, so, -20 m/s.0861

On impact, they stick together, so we are having a completely inelastic collision.0866

What is the velocity of the twisted hulk of metal they now make just after they collide?0870

We know that, p_sys(beginning) = p_sys(end).0875

p₁(initial) + p₂(initial) = p₁(final)+p₂(final).0886

What about the things on the right, we no more have two separate objects, we have one, stuck together.0900

In reality, this is not going to be that, it is going to the momentum of the twisted hulk of metal, momentum of the wreck, momentum of the whole.0911

So, m₁v₁ (initial) + m₂v₂(initial) = (m₁+m₂)v_hulk.0930

So, 1000×30 + 2000×(-20) = 3000v_hulk, so, -10000 kg m/s / 3000 = v_hulk = -3.33 m/s, approximately.0957

That is what is left, that is what the speed that is left, actually velocity, since we know that it is going to move in the direction of the heavier car, heavier car was going to slower, it had more momentum, so it winds out in the end.1000

We are going at a -3.33 m/s velocity.1013

Example 2: Two billiard balls of mass m₁ and m₂, same mass of 0.16 kg, are on a pool table.1020

One of them is moving horizontally with speed of v₁ = 5 m/s towards the other.1029

After the collision, the second ball is moving with a speed of v₂ = 3 m/s, at 30 degrees below the horizontal.1033

What is the first's velocity?1040

In this case, we need to figure out, how much is going to the right, and how much is going up.1042

Now, we can figure out what the momentum is for the first ball, the ball that is moving.1052

We can figure out, the second one, we know what the momentum in the second one is.1059

We can figure out how much it is moving this way, and how much it is moving this way, and once we know that, we can go back and we can figure out, in this one, these two added up, are going to have to wind up equaling this one's vertical motion, which turns out to be nothing.1062

And we know that these two added up, are going to have to wind up equaling the horizontal motion.1092

We are now going to start working on it.1104

Velocity for the ball that is first moving before impact is 5 m/s to the right, o m/s up and down.1111

v₂ = 3 m/s, 30 degrees below the horizontal.1119

Let us figure out what is v₂(final) in terms of component vector.1124

v₂(final) = cos(30)×3 m/s (since this side is adjacent), and the other side will be, sin(30)×3 m/s.1132

So, (2.598, -1.5).1157

That is an important thing that we have to catch specifically.1166

It is not just sin(30), it is actually, sin(-30), because it is below the horizontal, and we probably want to make going up positive, and going to the right positive, because that is what we are used to as standard.1168

So, it is not just sin(30), it is sin(-30) because it is below the horizontal, it is up to us to pay attention to the signs, we have to be understanding what we are saying with this.1181

So, (2.598,-1.5) m/s.1194

Now we know what the velocity of the second thing is.1198

From there, we can figure out what its momentum is.1200

First off, we know that the momentum at the beginning of the system is going to be equal to the momentum at the end of the system.1203

So, m₁v₁ = m₁v₁(after impact) + m₂v₂(after impact).1249

Remember, m₁ and m₂ are the same thing, so divide everything by that m₁, m₂ combination, since m₁ = m₂ = 0.16 kg.1275

So, velocity = v₁(final) + v₂(final), (5,0) = (v_1x,v_1y) + (2.598,-1.5).1292

At this point, we can add things together, so, remember, we are dealing with vectors, each one of the pieces of vector, x component is totally separate from the y component, so now we can break this into two separate equations.1335

We got, 5 = v_1x + 2.598, and, 0 = v_1y + (-1.5).1346

So, 1.5 = v_1y, and , 5-2.598 = 2.402 = v_1x.1362

Putting these together, v₁ = (2.402,1.5) m/s, and that makes good intuitive sense.1380

If you hit the ball, the other ball winds up moving away as well, you wind up getting something that is moving slower now.1399

If it is going down, if it was going horizontally first, and the other thing bounces down, it is going to have to get rebounded upwards.1405

So, it makes good intuitive sense, passes the sanity check, and works out.1413

We are able to start these things moving, because of conservation of momentum, and we can make it a little bit easier by canceling out the masses, because we know that they all have the same mass, so it is a little bit easier for us, and from there we break one of them into its component pieces, and from there we are able to work in factors, because vectors, we can just deal with the component pieces.1415

Example 3: Two rubber balls are on a flat surface.1434

The first one has a mass, m₁ = 2 kg, and the second has a mass, m₂ = 1 kg.1437

The first one is moving at a velocity 5 m/s.1442

The first ball collides with the second ball, and we are going to be able to treat this collision as elastic.1445

I know, normal everyday collisions do not wind up being elastic, they are almost all inelastic, but it is good for us to get some practice with an elastic one, and rubber balls do manage to keep a lot of their energy.1449

What is the velocity of each ball after the collision?1459

Notice, this ball is going to be moving, and this ball is going to be moving after the collision.1462

One thing to notice is, which ball has to be moving faster.1467

The ball on the right is moving slower than the ball on the left, that is not going to make sense, because that means that somehow collision happened and then this ball managed to move through the other ball.1471

That does not make sense, we know that these are solid objects that are moving in one direction, it cannot move through it, so that is going to guarantee the fact that the velocity of the second ball is going to have to be greater than the velocity of the first ball.1483

In this problem, we are also going to wind up ditching the vector notation on the top, because we are working only in one dimension.1496

It is little bit easier than normal , because we are only working in one dimension, and we got a whole lot of writing coming up to do in an elastic problem, so we are going to make it a little easier by ditching the arrows, but it is important to remember that we are working on the basis of the idea of vectors, we are still going to care about direction and all those things.1503

For the beginning, we know that v₂ > v₁.1525

Now, let us look at, what is momentum going to tell us about this!1530

We know that, p_sys(beginning) = p₁(final) + p₂(final), (one dimension, so ditching arrows on top, but all the ideas of vectors are still there, we really care about direction of momentum.)1532

m₁v₁ (initial) = m₁v₁(final) + m₂v₂(final).1584

So, 2 kg×5 m/s = 2×v₁ + 1×v₂.1617

Sp, 10 = 2v₁ + v₂.1630

Now, that i snot enough to solve, we got one equation and two unknowns, so we are not able to do it.1637

But luckily, this is an elastic collision, so now we can bring energy.1642

We know, Energy(initial) = Energy(final), and the final energy of the system, since one thing is moving at the beginning, it is just that one thing's motion.1645

The final energy of the system is going to be both the things moving together, so, E₁ + E₂.1655

What is the energy at the beginning? (1/2)×2×(v_i)² = (1/2)m₁v₁²+(1/2)m₂v₂².1663

We cancel (1/2)'s all the way across, substitute things in, 2×5² = 2v₁¹+v₂².1680

So, 50 = 2v₁²+v₂².1699

At this point, we have got two unknowns over in this one.1708

How do we do this? Go back, and substitute one equation into the other one.1711

We get, 10 - 2v₁ = v₂.1717

Move that over here, so, 50 = 2v₁²+(10-2v₁)².1721

Simplify that out, 50 = 2v₁²+100-40v₁+4v₁².1736

Rearranging, 6v₁²-40v₁+50 = 0. (quadratic).1750

If you are really good at factoring, you might be able to factorize this, but, most of us are not that great at factoring, so what do we use now?1770

We got two options: (1) calculator, graph it and see the roots, because we will be able to treat this as a single function and when that function equals zero, you would have found the answer to this equation.1776

Alternately, we could just throw down with the quadratic formula, and we will be able to figure out what it is.1789

What is the quadratic formula? (if you want to do advanced mathematics, or even physics, you could buy a graphing calculator, but it is not necessary.)1794

[-b +/- sqrt(b² - 4ac)]/2a.1821

Plug things in, [40 +/- sqrt(40² - 4×6×50)]/(2×6) = (40 +/- 20)/12, gives us two possible answers.1829

The two answers are, 60/12 = 5 m/s, and 20/12 = 1.67 m/s, approximately.1861

We have got two answers, so it is up to us to figure out which one makes sense, which one is going to allow v₂to be greater than v₁.1881

If it is 5 m/s for v₁, was it not 5 m/s beforehand?1887

That means that it hits the ball and then it keeps going at the same speed, that does not make any sense.1889

It manages to just go through the ball, remember, math is your friend, but it is not something that just spits out your answer.1905

It is up to you to keep it tamed, you have to understand what you are doing with it.1911

We brought this quadratic equation to bare, but it is up to us to understand how to interpret the results that it gives us.1915

It gives us two answers that work mathematically, the energy will be conserved, and the momentum will be conserved if that other ball was never there, it just manages to pass through it.1921

But, we know that it cannot be the truth, it did not just pass through it, it has to contact that ball.1929

So, the answer that has got to be, is this one tight here.1934

So we have got, v₁ = 1.67 m/s.1938

If that is the case, we can then plug this back in, and we get, 10 - 2×(1.67) = v₂, so, 6.66 m/s = v₂, approximately.1943

There we go, we are able to figure out what the other velocity has got to be, because we are able to bring all these equations bare.1964

We bring what we know about energy formula, what we know about momentum, and we are able to do both of them, because we got two unknowns and two equations, we are able to solve it.1971

And there are our answers.1979

We know that v₁ is 1.67 m/s, still in the positive direction, and v₂ is positive at 6.66 m/s, and it turns out it is going to be faster than the initial object actually was.1980

That is the stuff that happens in elastic collisions, things do not behave necessarily quite like you expected, because energy is conserved, so we got that v² thing going on, so some weird stuff can happen sometimes.1997

Well, not weird, because it is physics, it is reality, so it cannot be weird in that way, but it is certainly not what we necessarily expect.2008

Example 4: A ballistic pendulum is one way to measure the velocity of a bullet fired from a gun.2028

Ballistic pendulum is a big heavy thing sitting on a rod, and there is this thing here, and a bullet will come in here, and it is going to impart its momentum, so the thing will swing up.2032

It winds swinging up, just like we got here, bullet goes in here, launches itself in, and put some momentum into it, and the ballistic pendulum will continue to swing up.2050

It is one way to measure the velocity of a bullet fired form a gun.2061

A bullet is fired into a large mass at the end of a pendulum where it launched itself.2062

By measuring how high the pendulum rises, we will be able to figure out the velocity of the bullet before impact. How do we do that?2068

Denote the bullet's mass as m_b, the pendulum's mass as m_p, assume the pendulum's rod's mass is negligible, so all we have to worry about is the mass of the block.2074

The height that the block achieves will be 'h'.2086

Using all these things, (in real life, we could build something like this, and using a pencil we could mark how high it went up, and we will be able to weigh the mass of the bullet and the pendulum, we will be able to find a fairly light weight strong rod that was able to hold it in place without being much compared to the mass of the pendulum, and we will actually be able to do this, this is what people did in the 1700's).2091

What is the trick that we will be able to use to solve for this?2123

Momentum is not going to be conserved throughout the entire thing.2126

The momentum of the bullet, as it goes up, we have got gravity that is dealing against it.2130

So, gravity is an external force from the point of view of the momentum, if it is going up, it is going to have to have some force moving it up, so we have got external forces on the bullet.2135

But at the same time, energy is not going to be conserved.2147

We got this impact, we never said it was elastic, this is an inelastic collision, this is definitely an inelastic collision.2149

We cannot conserve energy throughout, we cannot conserve momentum throughout, how do we do this?2160

We can conserve them one piece at a time.2165

Notice, the instant, if we break this into three things, the bullet goes in, and at this moment, when it is just after the collision occurs, before it has the chance to swing up, momentum is conserved.2168

So, p_i = p_f, then from here, once it is swinging up, there is no more collisions, it is just a basic pendulum rod going up, so we can use energy.2187

So, for here, E_i = E_f.2199

We are able to break this into two pieces: the momentum is going to be conserved for just the collision, and after the collision, energy will be conserved.2203

We break it into two pieces: beforehand and afterwards.2212

If p_i = p_f, what is in the initial momentum?2215

m_bv_b, there is nothing else moving, we got the pendulum at rest.2222

Once again, we are going to nick those vectors, since we are dealing with linear quantities.2227

Wait a second! I hear you ask, is it not going to move up and to the right when it swings up!2237

Yes, that is true, but we will be dealing with that as energy, so when we get to that point, we only have to worry about speed.2242

For the collision, as far as we are concerned, it hits and goes in, and it is still basically moving in just one dimension, because it launches itself very quickly, and the collision happens very quickly compared to the movement of the swinging.2247

So, m_bv_b = (m_b+m_v)v_sys, for ease, we will denote, v_sys after collision as, v_p.2258

That makes it a little bit easier to write, because we are going to have to be talking about two very different world, the world of the impact, and the world after impact where it is swinging.2297

This will make it a little bit easier, v_p, it is important to pay attention to what wee are doing here.2303

We can figure out what the velocity of the bullet is, in terms of, (m_b+m_p)/m_b, but are not able to solve for this, we still need to figure out what the velocity of the pendulum is, we do not know what the velocity of the pendulum is.2309

Remember, we only have the height.2328

Now, start looking in terms of energy.2330

E_i = E_f.2332

The energy of the system at the beginning of the swing, does it have any height?2336

No, we can consider that as its base line height. Is it moving?2340

Yes, it is definitely moving, there are no spring involved, and the only thing that we got there is the kinetic mechanical energy, kinetic energy.2342

(1/2)(m_b+m_p)v_p² = Mgh (at the top of its arc, it has got to just finish moving, then the only thing it has is stored gravitational potential energy.)2349

'M' in this case, is not just 'M', it is the mass of the whole thing, m_b+m_p, so, m_b+m_pgh.2356

Cancel out m_b+m_p on both sides, so, v_p² = 2gh, v_p = sqrt(2gh).2387

Substitute that back in, v_b = (m_b+m_p)/(m_b×sqrt(2gh)), there we are.2402

By just measuring the height that the thing goes up to, by already knowing the constant of gravity, by weighing the mass of the bullet, weighing the block, we are able to figure out the velocity of the bullet, which is pretty great.2427

These are all the things that we were able to do in the 1700's.2437

Even without fancy technologies like LASER, and all these other cool stuff, we can actually come up with a really solid way to measure the velocity of a bullet by what we know about momentum and what we know about energy.2440

Hope you enjoyed this.2452