Video: Calculating Average Velocity Knowing Displacement and Time

An object is initially at rest and then moves 100 m to the left and then 40 m to the right. The time taken between the object first starting to move and having moved 40 m to the right is 20 seconds. What is the object’s average velocity to the left for the 20 s after it begins moving?

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Video Transcript

An object is initially at rest and then moves 100 metres to the left and then 40 metres to the right. The time taken between the object first starting to move and having moved 40 metres to the right is 20 seconds. What is the object’s average velocity to the left for the 20 seconds after it begins moving?

Okay, so in this question, we’ve got an object. So, let’s say this is our object. And we’ve been told that this object is initially at rest. And then, it moves 100 metres to the left. So, there it is, having moved 100 metres to the left. And then we’ve been told it moves 40 metres to the right. So, there’s the 40 metres to the right.

We’ve been told that the time taken between the object first starting to move and having moved 40 metres to the right is 20 seconds. In other words, it takes 20 seconds for the object to start moving, go 100 metres left, and then come back another 40 metres. That whole journey takes 20 seconds.

We’ve been asked to find the object’s average velocity to the left for the 20 seconds after it begins moving. But then we’ve just seen that the entire journey takes 20 seconds. And, of course, those 20 seconds are the 20 seconds after the object begins moving. So, in other words, we’ve been asked to find the object’s average velocity over its entire journey.

Now we can recall that the average velocity of an object is defined as the total displacement of the object divided by the total time of the entire journey. And this word, displacement, means the shortest distance between the start point and the end point. So, in this case, our start point is here, because that’s where the object began its journey. And the end point is here. Now the shortest distance between these two points is the distance in a straight line. Or, in other words, we’re looking for this distance here.

Let’s call this distance 𝑑. And since this distance is the shortest distance between the start point and the end point, let’s now also say that the displacement of the object is 𝑑. And, of course, because this is the start point and this is the end point, the displacement is actually going to be towards the left. Because, remember, displacement is a vector quantity. So, not only is it the shortest distance between the start point and the end point. But we also need to account for the direction. But anyways, so what is the value of 𝑑?

Well, we know that the entire distance that the object moves to the left is 100 metres. And then, it comes back 40 metres. And so, 𝑑 is going to be 100 metres minus 40 metres. And this ends up being 60 metres. And, of course, once again because displacement is a vector quantity, we need to say it’s to the left. In other words then, we could say that the object’s entire journey is equivalent to its starting here and just moving to the left until it reaches this point. That’s basically what the displacement is measuring.

And so, in order to find the average velocity of the object, we now know the total displacement. Which means we need to work out the total time of the Journey. But then, we’ve been given this in the question. We’ve been told that the entire journey lasts 20 seconds. And hence, we can say that the total time of the journey, which we’ll call 𝑡, is 20 seconds. So now, we can work out the object’s average velocity.

Let’s call the average velocity 𝑣 subscript avg, and say that it’s equal to the total displacement of the object divided by the time taken for the entire journey. Then we can sub in the values and say that 𝑣 subscript avg is equal to 60 metres to the left divided by 20 seconds. Now the only reason we’re including to the left, once again, is because velocity is also a vector quantity. And so, we do need to mention somewhere its directionality.

But anyways, so we can evaluate the fraction on the right-hand side of this equation. So, that’s 60 metres divided by 20 seconds. And this ends up being three metres per second, once again to the left. In other words then, even though our object started here, for example, and moved 100 metres left and then 40 metres to the right, we’ve just worked out that the average velocity of the object over its entire journey is basically equivalent to the object having moved at three metres per second to the left.

But then, in order to answer this question, we don’t actually need to mention “to the left”. Because in the question we’ve already been asked to find the object’s average velocity to the left. And so, we’ve arrived to our final answer. The object’s average velocity to the left for the 20 seconds after it begins moving is three metres per second.

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