Video: Determining the Average Velocity of an Object

An object moves north at 12 m/s for 10 seconds and then stops and stays motionless for 10 seconds before moving north at 12 m/s for another 10 seconds. What is the object’s average northward velocity?

06:24

Video Transcript

An object moves north at 12 metres per second for 10 seconds and then stops and stays motionless for 10 seconds before moving north at 12 metres per second for another 10 seconds. What is the object’s average northward velocity?

Okay, so in this question, we’ve got an object. Let’s say this orange blob is our object. We’ve been told that it moves northward at 12 metres per second for 10 seconds. Then we’ve been told that it stops and stays motionless for 10 seconds. And then finally, it moves north again at 12 metres per second for another 10 seconds.

So there are three stages to this object’s journey. The first stage is when it moves north at 12 metres per second for 10 seconds. The second stage is when it’s stationary for 10 seconds. And the third stage is when it’s moving north again at 12 metres per second for another 10 seconds. We’ve been asked to calculate the object’s average northward velocity.

Now first of all, let’s deal with the word northward. We already know that when the object does move, it moves north. This is true for both stage one and three when it’s moving. So we don’t need to worry about the word northward because when we calculate the object’s average velocity, automatically it’s going to be its northward velocity.

Now we want to calculate its average velocity. Intuitively speaking, we know that in its initial stage of motion, the first stage, the object is moving at 12 metres per second for 10 seconds. And then the object stops. And then in the third stage, it starts moving again. So the object’s average velocity is going to be somewhere between zero metres per second and 12 metres per second, because it spends some period of time traveling at 12 metres per second and some other period of time stationary as zero metres per second.

Now there’s two things we need to know in order to answer this question. Firstly, that the velocity of an object is defined as the distance covered by the object divided by the time taken for that object to cover that distance. And secondly, that the average velocity of an object, which we’ll call 𝑣 sub average, is equal to the total distance covered by that object, 𝑑 sub total, divided by the total time taken for the object to cover that distance, 𝑡 total.

Although if I had a journey as exciting as the object, I’d be 𝑡 total too. Badum tss!

Okay now, that was a terrible joke! Let’s move on from that. So what’s the difference between this definition and this definition? Well of course, this one deals with the average velocity, whereas this one is dealing with the velocity of the object in different stages of its motion. In other words, we can apply this expression separately to stage one to stage two and stage three of its travel, whereas this one can be applied to its entire journey to find out its average velocity.

Now we need to use this first expression to work out the distance covered in each stage of the object’s motion. To do this, we multiply both sides of the equation by the time 𝑡. What this leaves us with is that the velocity of the object multiplied by the time taken is equal to the distance covered. So we can work out the distance covered by the object in stage one, remembering of course that stage one is when the object moves north at 12 metres per second for 10 seconds the first time round.

Let’s call the distance covered in stage one 𝑑 sub one. Now 𝑑 sub one is equal to the velocity of the object during that stage of motion, which is 12 metres per second, multiplied by the amount of time for which it travels at this speed, which is 10 seconds. And so evaluating the right-hand side, we find 𝑑 sub one to be 120 metres, at which point we can move on to finding out 𝑑 sub two, the distance covered in the second stage of the object’s motion.

Now in the second stage, the velocity of the object is zero metres per second because it’s stopped; it’s stationary. The exact wording from the question is that it stays motionless. And we multiply this velocity by the amount of time for which it stays motionless, which is 10 seconds again. But zero times any other number is still zero. So the distance covered in the second stage of motion is zero metres.

This makes sense. The object is stationary; it’s not moving. So how can it cover any distance? Okay, so let’s move on to 𝑑 sub three. So after the 10 seconds during which the object is stationary, it starts moving again at 12 metres per second for another 10 seconds. So 𝑑 sub three is going to be the velocity, 12 metres per second, multiplied by the time for which it moves at this velocity, which is 10 seconds. And so we find 𝑑 sub three to be 120 metres.

Okay, so now we have the distance covered in the first stage of motion, second stage of motion, and third stage of motion. At this point, we can start working out the average velocity of the object, because if we have the three distances, we can work out the total distance covered by the object. This is because the total distance covered by the object, 𝑑 sub total, is simply going to be the distance covered in the first stage of motion, 𝑑 sub one, plus the distance covered in the second stage of motion, 𝑑 sub two, plus the distance covered in the third stage of motion, 𝑑 sub three.

And of course, we need to know what 𝑡 total is. This is the total amount of time taken for the object to move from start to finish. So this is equal to the time for the first stage of motion plus the time for the second stage of motion plus the time for the third stage of motion. So we can say that the average velocity of the object is equal to the total distance covered, which we have seen is 𝑑 one plus 𝑑 two plus 𝑑 three, divided by the total time taken, which is 𝑡 one plus 𝑡 two plus 𝑡 three.

At this point we can sub in the values. We have the distance covered in the first stage of motion, second stage of motion, and third stage of motion. And we’ve got the time taken for the first stage of motion, which is 10 seconds, the time taken for the second stage of motion during which admittedly it was motionless, which is also 10 seconds, and the time taken for the third stage of motion, which is another 10 seconds.

So we can evaluate this fraction to find, first of all, a total distance traveled by the object of 240 metres in the numerator divided by the total time taken, which happens to be 30 seconds. And then we simplify the fraction to give us an average velocity of eight metres per second. At this point, we found our final answer, and this kind of makes sense.

The object traveled at 12 metres per second for 10 seconds, then it stopped for another 10 seconds, and then it started moving again at 12 metres per second for a third set of 10 seconds. So the average velocity of the object is not going to be 12 metres per second because there was that chunk in the middle where it was stationary. Conversely, it’s not going to be zero metres per second because there was a chunk of time either side of it where it was moving at 12 metres per second.

The average velocity is going to be somewhere in between, and we found it to be eight metres per second. So we found our final answer: the object’s average northward velocity is eight metres per second.

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