Video Transcript
Calculate the average velocity of a car which moved in a straight line a distance of 120 meters with the velocity of eight meters per second and then moved in the same direction on the same line a distance of 180 meters with the velocity of six meters per second. Is it (a) seven meters per second, (b) 6.7 meters per second, (c) 6.3 meters per second, or (d) 7.2 meters per second?
So this question deals with a car moving in a straight line. The question doesn’t tell us which direction the car is moving in. So let us assume it is moving from left to right. This choice of direction doesn’t make a difference to the answer we will get, but it helps us draw what is happening. Let us assume that our car starts at this point here. And the question tells us that to begin with, the car moves 120 meters to the right to this point here with a velocity of eight meters per second. We are then told that the car moves a further 180 meters in the same direction along the same line, this time with a velocity of six meters per second. So overall our car has moved from its starting position here to its final position to the right over here.
Since the question asked us to find the average velocity of the car, we should begin by recalling 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. Here, let’s calculate the two quantities on the right-hand side of this equation separately; i.e., we will first calculate the total displacement of the car, and then we will calculate the total time of the journey.
Let’s start by calculating the total displacement of the car, where we recall that displacement is defined as the shortest distance from the start point of an object to the endpoint of an object. Remember that displacement is a vector quantity, so as well as having a numerical value, it also has a positive or negative sign, which tells us which direction the car is moving in. Here, we can assume that to the right is the positive direction. So any positive displacement means an object is moving to the right. For our car, the displacement is simply the straight line distance from the start point to the endpoint.
If we call the total displacement of the car 𝑑, then we can calculate 𝑑 by simply adding the distance that the car travels in the first part of the journey to the distance that the car travels in the second part of the journey. This means that 𝑑 is equal to 120 meters, which is how far the car goes in the first part of the journey plus 180 meters, which is how far the car goes in the second part of the journey. Now, if the second part of the journey was not in exactly the same straight-line direction as the first part of the journey, this displacement would’ve been more complicated to work out. But since they are in the same direction, we can simply add the two distances together.
So doing this addition, we see that the total displacement of the car is 300 meters. And remember that this displacement is a vector quantity. So the fact that it is positive tells us that the overall motion of the car is to the right. This means that we’ve done the first part of our question because we’ve calculated the total displacement of the car. So let’s write this displacement down here so we don’t forget it while we keep working on the rest of the question.
The second thing we need to work out is the total time of the entire journey of the car. The question doesn’t tell us how long the car is traveling, so we need to work it out from the information we do know. The equation we need in order to work out the time of the journey is that speed is equal to distance over time. We can rearrange this equation to make time the subject by first multiplying both sides of the equation by time, which effectively brings time up to the left-hand side of this equation. We can then divide both sides by speed, which effectively brings speed down to the bottom of the right-hand side of this equation. This means that our equation for time becomes time is equal to distance divided by speed.
Since the car traveled at two different speeds, we need to work out the time taken for each part of the journey separately because the equation we’ve just written down assumes that all of the numbers we are working with are constant. So let us denote the time taken for the car to complete the first part of the journey by 𝑡 one. Then from the equation we’ve just written down, 𝑡 one is equal to the distance traveled in the first part of the journey, which is 120 meters, divided by the speed that the car traveled at during the first part of the journey, which was eight meters per second.
We can work out the numerical part of this fraction, which is 120 divided by eight, to be 15. And the units which are meters divided by meters per second just give us second. So 𝑡 one, the time that the car takes to complete the first part of the journey, is 15 seconds.
Let us now call the time taken for the car to complete the second part of the journey, 𝑡 two. And we can work out 𝑡 two in the same way as we calculated 𝑡 one. That is, 𝑡 two is equal to the distance traveled in the second part of the journey, which is 180 meters, divided by the velocity of the car during that part of the journey, which is six meters per second. And if we work this out, we see that 𝑡 two is equal to 30 seconds.
Now we just need to calculate the total time of the journey. So if we call this total time 𝑡 with no subscript, then 𝑡 will just be equal to the sum of all the times of all the sub parts of the journey. That means that here 𝑡 is just equal to 𝑡 one plus 𝑡 two. So if we plug in the numbers we have just found for 𝑡 one and 𝑡 two, we see that the total time 𝑡 is equal to 15 seconds plus 30 seconds. So it is equal to 45 seconds. Let’s write this down to the side so we don’t forget it while we calculate the average velocity.
We now have everything we need to work out the average velocity of the car because we’ve worked out the total displacement of the car and the total time of the entire journey. So if we call the average velocity 𝑣 subscript avg, then from our definition of average velocity, 𝑣 avg is equal to the total displacement of the car divided by the total time of the entire journey, which we can write as 𝑣 avg is equal to 𝑑 divided by 𝑡. We then just need to substitute in the values we found for 𝑑 and 𝑡. We found that 𝑑 is equal to 300 meters and 𝑡 is equal to 45 seconds. So let’s write down that 𝑣 avg is equal to 300 meters divided by 45 seconds.
If we work this out, the numerical part of the velocity gives us 6.6 recurring, and our units are meters per second. This bar notation is how we write recurring decimals, which means that the six after the decimal point repeats forever.
This is close to our final answer. But if we look at all the multiple choice options we were given, we see that they are all given to one decimal place. So we should round our answer to one decimal place as well. If we round 6.6 recurring to one decimal place, we get 6.7. So we find the average velocity of the car to be 𝑣 avg is equal to 6.7 meters per second. Remember that, like displacement, velocity is a vector quantity. So the fact that this velocity is positive tells us that the car is moving to the right in this case. By comparing with our multiple-choice options again, we find a match, in this case (b). And hence, we can give this as our final answer that the average velocity of the car is equal to 6.7 meters per second to the right.
