High School Physics Linear Momentum

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

From our work and energy, we already know that the mass and the speed of an object is able to determine its kinetic energy.0006

But, when we were dealing with energy, we only had speed as the way of determining energy.0012

There was not anything talking about direction.0017

Kinetic energy was great for telling u slots of stuff, but it did not tell us if we were going to the north, south, up or down.0019

To capture that, we are going to introduce a new idea: Linear Momentum, what your motion is along a line.0026

We want the linear momentum to talk about an object's motion in a given direction, just like energy gave us an idea of speed and mass for an object, linear momentum will give us something that tells us about the motion of an object.0032

Consider two scenarios: We got a box moving at the same speed but in opposite direction.0045

To capture the difference, we will not be able to just use the speed, because it is moving in the speed in both the cases.0052

But we will need to also capture its direction.0057

To do that, we need to use vector.0060

We are going to have to use v, not just 'v', the speed, we need to have its actual velocity.0062

Also, what if the boxes had different masses?0071

If we had two different boxes, that were both moving in the same speed, one of them was say 1 kg, and the other a 100 kg, we probably want to think of them as being different things.0073

It will take a whole lot more effort to stop a 100 kg box than the 1 kg box.0083

So, direction is part of it, but we are also going to have to take into account, the mass.0088

Clearly, it is going to a good idea to include mass in our idea of linear momentum.0093

So, we are going to have to be able to deal with the velocity vector, not just the speed, and also mass.0097

Put in that together, we get, linear momentum, and we define it as, mass×(velocity vector), mv.0106

Notice that, since this is a vector quantity that we are dealing with, it is going to be, how much we are moving in the x coordinate, how much we are moving in the y coordinate, if you are also moving in the z coordinate, it is going to be, mv_x + mv_y + mv_z.0116

We are going to break it up as a vector, and m will just scale the vector.0135

These two characteristics, they define, what we are going to create as momentum.0138

Momentum, p = mv.0142

Why do we use a p?0146

I honestly do not have a good answer, I wish I did, there are possibilities, it has a Latin root, but I was not able to figure it out, sometimes there are mysteries in the world.0148

An important thing to notice here is that, this p is not just a scalar quantity, it is not just a single number, it is a vector.0166

If v comes in (x,y,z), our p is also going to have to come in (x,y,z).0174

Units of linear momentum are, mv, so kg×(m/s).0183

We are going to consider a new idea: Impulse.0191

What if we want to talk about how much an object's momentum changes, that is important.0193

If we got a box moving along, and if we put a force on the box, we are going to change the momentum of the box, because we will change the speed that it is moving at.0198

So we are going to define the idea of impulse.0205

What really changes the velocity of the thing?0207

It is just going to be the fore involved, but not just the force.0212

The same force is going to be very different if you put a 100 N on an object for 0 s, 1 s, 10 s, 100 s, totally different things are going to happen depending on the amount of time that the force is acting on it.0217

The objects mass remains constant, pretty reasonable.0227

The object's velocity, and thus its momentum, is going to change based on the force applied, and how the long that force lasts.0232

We define impulse as the letter 'j' (no particularly good reason here, just making sure we are using letters that have not already been taken by somebody else), j= force×time = Ft, just like before, j is a vector because force is a vector.0238

Makes sense, because we are talking about change in a vector quantity.0255

Note that impulse is a vector, and its units are going to be, Ft, so N s.0259

At this point, we have created some definitions, and we can see that linear momentum and impulse are connected because we wanted impulse to represent a way of shifting around momentum.0268

You put force into an object for a certain object of time, it is going to change the momentum that the object has, because we will be changing the speed that it is changing at.0279

But, what is the precise mathematical relationship?0286

Let us figure it out.0288

We look more closely at the formula for impulse.0289

j = Ft, if we expand that out, we can get this.0292

F = ma, so, j = mat = mΔv, (since 'a' is how much your velocity is changing with time).0299

Since mass is not changing, we can pull that change outside, and we get, Δv, because we do not have to worry about, since velocity is the only thing that can change, we are assuming that the mass is constant, so, mΔv is the same thing as, Δ(mv), because mass is just a constant, and velocity is the variable, at this point.0328

Remember, we defined, p = mv, so it has to be the case that, Δ(mv) is the same as, Δp.0350

So, in the end, j = Δp, so impulse is simply the change in the momentum.0362

Note that impulse and momentum have the same units, 'N s' is the same thing as 'kg m/s', because N comes from, if F = ma is kg m/s/s, so, N is kg m/s/s, and multiply with s, so kg m/s, is what we had for momentum.0371

It makes a lot of sense, our units wind up working out, so, j = Δp.0404

In this section, we have talked about what linear momentum is, but why have we talked about 'linear' momentum, when we have not heard about any other kinds of momentum, why is it called linear momentum if it is the only momentum that we are concerned with?0417

The thing that is going on, is there are other kinds of momentum, there is angular or rotational momentum.0438

Spinning objects, objects that are spinning, if you take a wheel and you spin it really fast, it will keep spinning, right?0445

It has a momentum, it is not moving anywhere, it is just sitting there in space and spinning, but it takes effort to start it spinning, and it takes effort to stop it spinning, so there are torques involved, we have not talked about rotational mechanics.0452

The entire thing in Physics you cannot talk about, but we just do not have quite enough math to really feel comfortable handling it, we are almost there, this is definitely close to being within our grasp, but a little too much math for us to tackle there, so we are holding off on it, that is why we have not talked about angular momentum, which is also similar to rotational momentum.0465

We have been talking about linear momentum, because we want to make sure that this is kept clear, as this is linear momentum as opposed to this other kind of momentum, but often when we are talking about linear momentum, we will also just refer to it simply as momentum, because that is the thing that is more common, but it is important to keep in mind that there are other kinds of momentum.0485

Just because we are talking about one of them, does not mean that there is nothing else out there.0501

Let us start with our examples.0506

A skate board of mass 4 kg is rolling along at 10 m/s.0508

What is its momentum? This one is pretty easy.0512

What is the basic definition for momentum? p = mv = 4 kg × 10 m/s = 40 kg m/s. (since one dimension, velocity becomes a single number).0513

Same skateboard, m = 4 kg, is rolling along with an initial velocity of 10 m/s, just like before.0538

A force of F = -6 N is applied to it, for t = 6 s.0547

At the beginning of this problem, we got some skateboard rolling along on the ground, and it is moving this way.0551

However, as time moves on, there winds up being a force applied to it in this direction, so later on, this skateboard is going to be rolling along with a much smaller velocity vector.0558

It is still going to be moving forward, potentially, depends on how long that force is actually, may be that force is going to push it so hard that it winds up going in the other direction, we are going to have to do some math to figure it out.0569

But, the force is acting on it in the direction opposite of current travel, we are travelling in the positive direction (right), and now force is going to wind up in the negative direction.0583

That is the importance of using vectors, we know how positive and negative direction, even if we are still on one dimension.0596

So, What is the impulse vector?0602

Impulse, j = Ft = -6×6 = -36 N s, so what is the final velocity that it is going to have?0604

In this case, we know that change in momentum is equal to the impulse.0629

We already figured out what the initial momentum is.0635

In the last problem, it wound up being, 4×10 = 40 kg m/s, so the final one is going to be, p_i + j = p_f, since p_f - p_i = j.0638

p_i = 40, and the change is -36, so in the end we get, 4 kg m/s = p_f, is the final momentum.0670

Final momentum does not quite tells us the final velocity.0684

But we can figure that out pretty easily from there.0687

p_f = mv_f, 4 = 4v_f, so, v_f = 1 m/s, is the final velocity.0688

It is still moving in the positive direction.0709

It would be possible to figure this out without using momentum and impulse, but momentum and impulse may dissolve pretty simple things that we have to do, very direct, multiplying and then adding, and then doing some really simple algebra.0711

but, we could go back to doing this with Newton's second law.0725

If we want to do this in Newton's second law, we have got, F = ma, -6 = 4a, a = -1.5 m/s/s.0729

What does the change in velocity wind up being?0753

Δv = -1.5×t = -1.5×6 = -9 m/s, and so, if you started with initial velocity of 10 m/s, then, v_f = 10 + (-9) = +1 m/s.0755

So, if we wanted, we could do this in terms of basic fundamental, Newton's second law, but in this case it is pretty easy.0780

Remember, the way momentum works, the way we have defined it, is really just sort of jumping off the point of using Newton's second law.0789

That is how we got impulse.0796

Impulse was based around the fact that, the reason why impulse is equal to the change in momentum is because we used F = ma, at what point we got, Ft, so in the end, they are deeply interconnected.0797

So, we can decide to go with Newton's second law, but in lots of problems, it is going to wind up being the case that it is actually a little bit easier the way of linear momentum, especially when we are dealing with momentum problems.0812

In the next lesson, we are going to wind up seeing why it is really useful to have momentum when we get to the conservation of momentum, and that is why this stuff really matters.0821

Example 3: A ball of mass m = 0.5 kg is moving horizontally with v_i = 10 m/s.0835

It bounces off a wall, after which it moves with v_f = -7 m/s.0842

What is the change in linear momentum, what is the magnitude of the impulse?0848

We got this ball, moving along the positive direction, and it hits the wall, and afterwards it changes, rebounds, and it is moving in the negative direction.0851

What is the initial momentum? mv = 0.5×10 = 5 kg m/s.0862

What is the final momentum? 0.5×(-7) = -3.5 kg m/s.0885

So, the change is, final - initial, Δp = -3.5 - 5, seems a bit weird, but makes sense, we started off in the positive direction, and ended up going in the negative direction, so the entire change has got to be one of negative momentum occurring.0898

So, -8.5 kg m/s is the change in the linear momentum.0937

The magnitude of impulse, remember, the magnitude is the size of something, so the size of this, j = Δp = magnitude(-8.5 kg m/s) = 8.5 kg m/s.0945

In the end, when we are dealing with magnitude, it does not care about direction, it does not care about positive or negative, it just cares what [unclear] like the thing we are dealing with.0972

In this case, we had -8.5 as the change in momentum, but the magnitude of the change in momentum was just the total moment, 8.5.0983

This is sort of similar to what we saw in energy before, instead of using velocity, it was the length of the velocity vector that we cared about, its speed.0993

It did not matter if it was pointing flat, it was pointing straight up , pointing at a 45 degree angle, all that mattered was what the total length was.1001

That is what we are seeing here, when we ask what the magnitude of the impulse, we are asking for what is the length of that thing.1010

In multiple dimensions, we take, sqrt(x²+y²+z²), because that is how we take the magnitude of a vector.1017

Last example: A ball of putty with a mass of 1 kg is about to fall on your head.1027

The velocity initially is -5 m/s, makes sense since it is falling down.1032

Which one is going to hurt worse, it lands and sticks on your head, so it lands and sticks, so the final velocity is zero, or, it lands and bounces off your head, with a final velocity of 4 m/s.1037

Let us call these cases, A and B.1048

In case A, it hits your head, and it sticks in place.1052

In case B, it bounces off your head.1059

Which one of these is going to hurt worse?1062

Normally we think of things being bouncy as good, they are easy right?1065

A super ball being bounced on your head, and bouncing off your head, it would not hurt much.1069

But what is really going on, which one is taking more force, the super ball that hits your head and just sort of sits there, or the ball that hits your head and ricochets right off?1075

We can break this down with Physics and Math.1083

In both cases, what is really going to be defined as hurting?1086

The way that we define as 'hurt' is probably the amount of force that it exerts on you.1092

So, you would rather have 2 N of force shoved on you than a billion newtons of force being shoved on you, a billion newtons force applied on you, your body is going to look like a pancake.1097

But 2 N of force, a small amount of force is going to hurt less than large amounts of force.1108

What we are looking for is, which one of these cases is going to produce less force on impact.1115

In both of these cases, it is going to be important to know how long the balls contacting your head, we will be able to figure out what average force is going to be.1121

Impact time for both of these cases will be 0.25 s, so let us figure out how much force is involved.1128

To do that, we need to figure out what the impulse in both cases are.1135

To do that, we need to know what the initial velocities are, what the initial momentum is, and what the final momentum is.1139

From there, we will be able to figure out the impulse, and from impulse, we will be able to figure out pretty easily what the force is.1145

In both them, we need to know what is their initial momentum.1150

Initial momentum = m×v_i = 1×(-5) = -5 kg m/s.1154

Case A, it lands and sticks on you head, so its v_f = 0.1168

If that is the case, what is the final momentum?1174

Still mass of 'm', but now it sticks, so it now has no velocity, so it has got a momentum of zero in the end, 0 kg m/s.1177

Compare that to B, where we have got a final momentum = 1×4 = 4 kg m/s.1189

It might make good sense at this point to go, 'Okay, the one with less momentum is case B, because we go from 5 to 4'.1207

But that is not the case.1215

The whole case is, we go from -5 to 4.1217

So which is a bigger change, going from -5 to 0, or -5 to 4?1220

We check that out, we get, Δp = (final) - (initial) = +5 kg m/s, is in A.1224

In B, Δp = 4 - (-5) = 9 kg m/s.1240

So, there is more change in momentum in case B than in case A.1254

So it is going to make more sense for B to hurt, because that force, to change that momentum, has to come from somewhere.1257

We have got a constant time, so it has got to be, the amount of force involved is going to have to be more in case B.1264

We will finish this out, at this point we can see that the answer is going to be case B.1272

So, j = Ft, we also know that j = Δp, so for case A, we got, 5 = F×0.25 s, F = 20 N.1276

So, 20 N is how much force you wind up undergoing for A.1301

Now, how much force does you wind up undergoing in B?1306

j =Ft = Δp, so, 9 = F×0.25, so, F = 36 N, in part B.1310

So, part B winds up putting more force on your head, in both cases, they are pretty small, so worst case you are going to have a little bit of headache, but 36 N is more force you have to suffer than 20 N.1328

So, it is actually better to have an object that lands on your head and just sticks there, something that goes 'splat!', than something that goes, 'boing!', because 'boing!', that force to make it bounce off is going to have to come from somewhere.1340

It is coming from your head, that is going to make it hurt more.1351

It is better to have it land and splat, it has less force, because it has to change its momentum if it is going to be able to bounce off your head.1354

I hope this lesson made sense, I hope you got a good understanding of linear momentum, because the next thing that is coming is conservation of momentum, and that is where the real point of momentum is.1361

Alright, good day!1368