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Question Video: Solving Word Problems Involving Average Velocity Mathematics

A person is late for an appointment at an office that is at the other end of a long, straight road to his home. He leaves his house and runs towards his destination for a time of 45 seconds before realizing that he has to return home to pick up some documents that he will need for his appointment. He runs back home at the same speed he ran at before and spends 185 seconds looking for the documents, and then he runs towards his appointment again. This time, he runs at 5.5 m/s for 260 seconds and then arrives at the office. How much time passes between the person first leaving his house and arriving at his appointment? What is the distance between the person’s house and his office? What is the person’s average velocity between first leaving his house and finally arriving at his office? Give your answer to two decimal places.

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Video Transcript

A person is late for an appointment at an office that is at the other end of a long, straight road to his home. He leaves his house and runs towards his destination for a time of 45 seconds before realizing that he has to return home to pick up some documents that he will need for his appointment. He runs back home at the same speed he ran at before and spends 185 seconds looking for the documents, and then he runs towards his appointment again. This time, he runs at 5.5 meters per second for 260 seconds and then arrives at the office.

We’re then asked three different questions. So let’s begin with the first one. How much time passes between the person first leaving his house and arriving at his appointment?

It might be helpful to begin by visualizing what happens at each stage of this person’s journey. The journey begins with running for 45 seconds towards the office. The person then realizes that they’ve forgotten something they need, so they go home at the same speed as they traveled before. So that means that the time will also be 45 seconds. They then spend 185 seconds looking for these documents but not traveling anywhere. And then, finally, he runs towards the office at 5.5 meters per second for 260 seconds.

We’re then asked for the time that passes between first leaving and then arriving at the appointment. So that means that we just add up the four time periods: 45 seconds, 45 seconds, 185 seconds, and 260 seconds. And when we work that out, we get 535 seconds. And that’s the answer for the first part of this question.

The next question asks us, what is the distance between the person’s house and his office?

In order to find the distance, we can use this information on the last stage of the journey, when we’re given the speed and the time taken. We can remember that distance is equal to speed multiplied by time. So we can fill in the values then. The speed is 5.5, and the time is 260. It is always worthwhile making sure that we do have the same equivalent units. In each case, the time unit is given in seconds, so we can simply multiply these values. When we work this out, we get a value of 1430. And the units here will be the distance units of meters. And that’s the second part of this question answered.

The third part of this question asks, what is the person’s average velocity between first leaving his house and finally arriving at his office? Give your answer to two decimal places.

We can recall the formula that average velocity is equal to net displacement over total time. In this problem, the net displacement will simply be the direct distance between the man’s home and the office. We have already calculated this distance in the second part of the question. It’s 1430. And the total time taken in the whole journey was 535 seconds. This gives us 2.672 and so on. And when we round that to two decimal places, we have a value of 2.67 meters per second. So if the positive direction is from home towards the office, then the person’s average velocity can be given as 2.67 meters per second.

It’s worth noting that if we’d been asked for the average speed instead, we would’ve needed to know the distances in the first two parts of the journey along with the distance in the final part of the journey. In this case, average speed would have been calculated by the total distance divided by the total time. However, since average velocity uses displacement, then we have the value of 2.67 meters per second.

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