Video: Determining Whether a Quantity Is a Vector or A Scalar from The Quantities Used to Calculate It

If a force is divided by a time, is the resultant quantity a vector quantity or a scalar quantity?


Video Transcript

If a force is divided by a time, is the resultant quantity a vector quantity or a scalar quantity?

Okay, so to answer this question, let’s first think about force and time, individually. Let’s recall that if we have some sort of object and we apply a force to that object, so let’s say we apply a force with magnitude 𝑓, then the force acts to try and accelerate the object. In fact, if the force 𝑓 is the only force acting on the object, then we can use Newton’s second law of motion to tell us that the force will accelerate the object in the direction of the force.

And so because of all of this, it’s important to know which direction the force is acting in. And hence, force is a vector quantity, where we can recall that a vector quantity is one which has magnitude or size, as well as direction in space. And when we say space, we don’t mean like outer space, we mean in three-dimensional space. For example, the force 𝑓 is acting towards the right as we’ve drawn it. And hence, we can say that that particular force has a magnitude of 𝑓 and a direction in space, which is towards the right. Hence, forces are vector quantities. So in our question statement, we can label force as a vector.

However, time, on the other hand, is a scalar quantity, where a scalar quantity is only one which has magnitude. It does not have a direction in space. Now, we could say that time flows forward or backward. And, those are the two, quote, unquote, directions of time. But when we’re talking about vector quantities, we’re talking about directions in space. And in particular, time cannot point towards the right, for example, or towards the left. And hence, time is a scalar quantity.

Now, what we’re trying to work out here is whether the quantity that we get by dividing a force by time is a vector quantity or a scalar quantity. So just for our convenience, let’s say that this quantity 𝑓 divided by 𝑡, force divided by time, is called 𝑥. It doesn’t matter what 𝑥 is, but it’s basically a force per unit time or force divided by time. And as we’ve seen already, the force is a vector, whereas time is a scalar. Now when we divide a vector quantity by a scalar quantity, that’s essentially equivalent to just rescaling the magnitude or size of the vector itself.

For example, in this particular case, we could say that a force of 40 newtons was applied over a time of five seconds. And so, we could work out the average force applied per second, which would be 40 newtons divided by five seconds. And so, that quantity would be eight newtons of force applied per second.

However, we can see here that this quantity does not lose its directionality. In other words, the original force that we applied over a period of five seconds was 40 newtons. But, we’ve drawn an arrow to represent that that force was acting towards the right. And so, applying 40 newtons over five seconds is equivalent to applying eight newtons every second, but in the same direction. And hence, this quantity 𝑥, which is the force divided by the time, also has a direction. It is a vector quantity. And so, our answer to this question is that the quantity found by dividing a force by time is a vector quantity.

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