Lesson Video: Momentum Physics • 9th Grade

In this video, we will learn how to use the formula for momentum, 𝑝 = 𝑚𝑣, to calculate the momentum of objects and calculate changes in an object’s momentum.

14:38

Video Transcript

In this video, we will be discussing a property of objects that have mass and are moving. This property is known as momentum. So, let’s first start by understanding what momentum is. If we start by thinking about an object — let’s say this ball here — and we say that the ball has a mass 𝑚 and is moving, in this case, toward the right at a velocity 𝑣. Then the momentum — which we will call 𝑝 — of this ball is defined as the mass of the ball multiplied by its velocity. This is a general definition of momentum, the mass of an object multiplied by the velocity with which it’s moving.

So, let’s now imagine that our ball over here has a mass 𝑚 of two kilograms and a velocity 𝑣 of three meters per second. In that case, we can say that the momentum of this ball 𝑝 is equal to the mass, which is two kilograms, multiplied by its velocity, which is three meters per second. This gives a numerical value of two times three, which is six, and the unit is going to be kilograms times meters per second. So, we find that the momentum of the ball 𝑝 is equal to six kilogram meters per second. We use this dot here to signify multiplication. In other words, we’re multiplying the units of kilograms by the units of meters per second. So, this is how we calculate the momentum of an object. We multiply its mass by its velocity.

And we’ve also seen that the units of momentum, at least in SI base units, are kilograms meters per second. However, it’s worth noting that we may see momentum written in other units, units such as grams multiplied by meters per second, or maybe kilograms multiplied by kilometers per hour, and so on and so forth. As long as we’ve got a unit of mass multiplied by a unit of velocity, the overall unit is going to be one of momentum. And in base units, that happens to be kilograms multiplied by meters per second.

So, let’s now imagine the following scenario. Let’s say we’ve got this blue ball here with a mass 𝑚 and a velocity 𝑣. We’ve already seen that the momentum of this ball is going to be equal to its mass multiplied by its velocity. And let’s now imagine that we’ve got another ball, this time a pink one, which has the same mass as the blue ball but a larger velocity. We’ll call this capital 𝑉. In this case, using the momentum equation, we can see that the momentum of the pink ball is equal to the mass of the pink ball, lower case 𝑚, multiplied by its velocity, uppercase 𝑉.

And since we’ve said that the velocity uppercase 𝑉 is larger than the velocity lowercase 𝑣, this means that the momentum 𝑚 multiplied by uppercase 𝑉 is larger than the momentum 𝑚 multiplied by lowercase 𝑣. And so, what we’re saying here is that the pink ball, which has the same mass as the blue ball but is moving at a larger velocity than that of the blue ball, also has a larger momentum than the blue ball.

Additionally, if we now think of a different ball, this time having a mass capital 𝑀, which is larger than the mass of the blue ball but it has the same velocity as the blue ball, lowercase 𝑣. Then, we can say that the momentum of the orange ball is equal to the mass, uppercase 𝑀, multiplied by its velocity, lowercase 𝑣. And once again, we see that, because the mass of the orange ball is larger than the mass of the blue ball, even though they’re traveling at the same velocities, the momentum of the orange ball is larger than the momentum of the blue ball. So, what we’ve learnt is the following. For two objects with the same mass, the one with the larger velocity will have the larger momentum. And also, for two objects with the same velocities, the one with the larger mass will have the larger momentum.

But this logic doesn’t necessarily just have to apply to different objects. That is, if we were to now just think about the blue ball and ignore the pink and the orange balls, we would see that if the velocity of the blue ball were to increase to, say, capital 𝑉, then the momentum of the blue ball would also increase to 𝑚 multiplied by capital 𝑉. And so, an object’s momentum can change over time. If its velocity increases, its momentum increases as well and vice versa. And this is also true for mass. For example, our ball could be rolling on some flat ground. And as it rolls, it picks up some dust, so its mass increases. Well, if its mass increases to, let’s say, capital 𝑀 and still moving at a velocity capital 𝑉, then the momentum of our ball, 𝑝, will be capital 𝑀 multiplied by capital 𝑉. And so, what we’re seeing here is that if an object’s mass or velocity changes over time, then its momentum also changes over time.

Now, here’s something interesting. Let’s notice that, in the equation for momentum, we’re using the velocity of an object and not its speed. We can recall that the velocity of an object is a vector quantity. This means that it has both magnitude, or size, and direction as well. So, if we return to the blue ball that we saw earlier, it’s important to say that the velocity of this blue ball is 𝑣 toward the right. In other words, we include the magnitude, that’s 𝑣, and the direction toward the right. And so, we’ve seen that velocity is a vector quantity. Additionally, the mass of an object is a scalar quantity. It only has magnitude and not direction because we don’t talk about the mass of an object in a particular direction. It just has this mass.

And so, in order to get momentum, we’re multiplying a scalar quantity by a vector quantity. And this means that momentum itself must also be a vector quantity. It must have magnitude and direction as well. After all, the velocity of an object is a quantity that contains information both about the speed of the object and the direction in which it’s moving. So, when we multiply this velocity by a mass, we get momentum. And therefore, momentum must also contain information about the direction in which that object is moving. We therefore say that an object which has a mass 𝑚 and is moving toward the right with a velocity 𝑣 has a momentum 𝑝, which has a magnitude or size of 𝑚 multiplied by 𝑣. And crucially, this momentum is to the right, in other words, in the same direction as the velocity.

So, the whole point of this is for us to remember that when we calculate an object’s momentum, we can use 𝑝 is equal to 𝑚𝑣 to calculate the momentum’s magnitude or size. And we also separately have to account for the direction of the momentum. This becomes most useful when we start thinking about the momentum of not just one object but the total momentum of more than one object. Let’s imagine that we’re now thinking about two balls, a blue one and a pink one. Let’s say that both have a mass of one kilogram each, and that both are moving toward the right with a velocity of one meter per second each. In this case, what’s the total momentum of both of these balls combined? Well, we can say that this total momentum — we’ll call this 𝑝 subscript tot — is equal to the momentum of the blue ball — we’ll call this 𝑝 subscript b — plus the momentum of the pink ball — we’ll call this 𝑝 subscript p.

And then, we can recall that the momentum of any particular object is equal to its mass multiplied by its velocity. And so we can say that the momentum of the blue ball is equal to its mass, one kilogram, multiplied by its velocity, one meter per second. And to this we added the momentum of the pink ball, which is also one kilogram — which is its mass — multiplied by its velocity of one meter per second. For the momentum of the blue ball, the numerical value becomes one times one. And the unit, of course, is kilograms times meters per second, or kilogram meters per second. Which means that the momentum of the blue ball is one kilogram meter per second. And here, we’re doing exactly the same calculation for the pink ball. And so, the momentum of the pink ball is one kilogram meter per second as well.

This means that the total momentum of the two balls combined is equal to one kilogram meter per second plus one kilogram meter per second. Now, one plus one is two. And since we’ve got the same units for both quantities, this means we can add them together. This gives us a total momentum for both the blue and the pink balls combined of two kilograms meters per second. Now as an aside, by the way, the reason we find the total momentum of the blue and the pink balls combined is because sometimes we need to consider the effect of the motion of multiple objects. For example, instead of two balls, we may’ve had an orange object that was made up of two components, the blue one and the pink one. And the overall motion of our orange object was dependent on the motion of the blue and the pink objects. So, that’s one reason we may have to find the total momentum of more than one object combined.

Anyway, so in this case, we found that the total momentum of two objects combined, the blue ball and the pink ball, as two kilograms meters per second. However, let’s now imagine the following scenario. Let’s imagine that the blue ball stays exactly as it is, a mass of one kilogram and a velocity of one meter per second to the right. Let’s also imagine that the pink ball has the same mass as before of one kilogram, but this time is moving toward the left at one meter per second. This is where we need to be careful about the vector nature of both velocity and momentum. Because now that we’ve got velocities of two objects in opposite directions, one is moving to the right and the other is moving to the left, we need to account for this by saying that one of these velocities is positive and the other is negative.

So, for argument’s sake, let’s just say that we choose that any velocity toward the right is positive and any velocity toward the left is negative. This means that our blue object is moving to the right with a speed of one meter per second and has a velocity of one meter per second to the right and hence has a velocity of positive one meter per second. Whereas our pink ball has a speed of one meter per second as well, but a velocity of one meter per second to the left or, in other words, a velocity of negative one meter per second. Therefore, when we go to find the total momentum of both the blue and the pink balls combined. In this particular case, we find that the momentum of the blue ball is equal to its mass, one kilogram, multiplied by its velocity of one meter per second, since it’s positive, plus the momentum of the pink ball. Which is its mass, one kilogram, multiplied by its velocity, which is now negative one meter per second.

And so, we find that, for the blue ball, we’ve got one kilogram multiplied by one meter per second, which is simply one kilogram meter per second. And for the pink ball, we’ve got one kilogram multiplied by negative one meter per second, which is simply negative one kilogram meter per second. And so, in order to find the total momentum, we’re adding one kilogram meter per second to negative one kilogram meter per second. These quantities add together to give zero. And so, we find that the total momentum of the orange object consisting of the blue ball and the pink ball is actually zero kilogram meters per second, which actually makes some sense when we think about it.

In this particular instance, we’ve got a blue ball which has a momentum of one kilogram meter per second to the right. And that momentum, in this case, is perfectly canceled out by the momentum of the pink object, which is one kilogram meter per second to the left or, equivalently, negative one kilogram meter per second to the right. Since we said that anything toward the right is positive and anything toward the left is negative. And because the momenta of the two objects cancel out, the momentum of the system, as we call it, which consists of the blue and the pink ball, is zero kilogram meters per second. So, we found a special case where the total momentum of our system, the orange object, is zero because the momenta of its components cancel each other out. And this is where the directionality of both velocity and momentum are really important. In other words, the fact that they are vector quantities is really important.

So, now that we’ve understood a bit about what momentum is and how we can apply it to certain objects, let’s take a look at an example question.

A cat has a mass of three kilograms. The cat moves four meters in a straight line in a time of two seconds. What is the momentum of the cat?

Okay, so in this question, we’ve been told that we’ve got a cat which has a mass, which we will call 𝑚, of three kilograms. We’ve also been told that the cat moves four meters in a straight line in a time of two seconds. So, let’s say that the cat starts here and ends up here. And we can say that the distance moved by the cat is four meters, and it does this in a time of two seconds. Based on this information, we’ve been asked to find the momentum, which we will call 𝑝, of the cat. So, let’s first start by recalling that the momentum of an object 𝑝 is defined as the mass of that object multiplied by its velocity. So, if we want to find the momentum 𝑝 of our cat, we need to know its mass and its velocity. Well, we already know its mass. We know that it’s three kilograms.

However, we haven’t been given its velocity in this question. What we’ve been given, instead, is enough information to calculate this cat’s velocity. Let’s recall that the velocity of an object is defined as the displacement 𝑠 of the object divided by the time taken for the object to travel that displacement. Now, the displacement 𝑠 of an object is simply the distance between its start point and its finish point in a straight line. And luckily, we’ve been told that the cat moves in a straight line. Therefore, in this particular case, the cat’s displacement is four meters. And the time taken to move that displacement 𝑡 is two seconds. Hence, we can say, first of all, that the velocity of our cat is equal to the displacement, which is four meters, divided by the time taken to travel that displacement, which is two seconds. And this gives us a numerical value of four divided by two and a unit of meters divided by seconds or meters per second.

And so, we find that the velocity of our cat is two meters per second. And because velocity is a vector quantity — which, in other words, means that it has magnitude and direction — we can say that the velocity is in the same direction as the cat’s displacement, which is also a vector quantity. In this case, we’ve drawn it as moving toward the right. However, we haven’t actually been given this information in the question. We just sort of assumed that the cat was moving toward the right when we drew our diagram. So here, we don’t need to worry too much about the direction in which the cat is moving. And therefore, we don’t need to worry about the direction of the cat’s velocity. All we care about is that the cat’s velocity has a magnitude of two meters per second.

This means that we can now calculate the cat’s momentum 𝑝, which happens to be the mass of the cat — which we know is three kilograms from the question — multiplied by its velocity — which we’ve just calculated to be two meters per second in the line above. And so, we find that the momentum 𝑝 of the cat has a numerical value of three times two, which is six, and a unit of kilogram times meters per second, which is kilogram meters per second. Hence, our answer is that the cat has a momentum of six kilogram meters per second. And once again, even though momentum is a vector quantity — that is, it has magnitude and direction — because we haven’t explicitly been told the direction in which the cat is actually moving, we don’t need to give the direction in our answer. We can simply say that our cat has a momentum of six kilogram meters per second.

So, now that we’ve had a look at an example question, let’s summarize what we’ve talked about in this lesson. Starting off, we saw that the momentum, 𝑝, of an object with a mass 𝑚 and a velocity 𝑣 is given by 𝑝 is equal to 𝑚𝑣. Secondly, we saw that momentum is a vector quantity. This means that it has magnitude, or size, and direction. And finally, we also saw that momentum has units of kilograms meters per second in SI base units, but can also be written in grams meters per second or kilograms kilometers per hour and other units of mass multiplied by velocity.

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