Two joggers start at the same time from opposite ends of an...

Two joggers start at the same time from opposite ends of an 8 -mile jogging trail and begin running toward each other. One jogger is running at a rate of $5 \mathrm{mph}$, and the other jogger is running at a rate of 7 mph. How long, in minutes, after they start will the two joggers meet?

Key Concepts

Unit Conversion

In many motion problems, the final answer must be expressed in specified units, such as minutes. Converting units—like converting hours to minutes by multiplying by 60—is necessary to ensure that the solution is presented in the correct and easily understandable format.

Relative Speed

When two objects move towards each other, their speeds add together to form the relative speed. This combined speed represents how quickly the distance between them is being closed, which is crucial for determining the meeting time in problems involving two moving entities.

Distance-Rate-Time Relationship

This concept involves the formula distance = rate × time, which is fundamental in solving motion problems. It is used to calculate the distance covered by an object when its speed and travel time are known, or to find the time required to cover a given distance at a specified speed.

Transcript

00:01 Here we have a problem where there's two joggers who are running towards each other at different speeds.

00:05 So they're on different ends of this path, which is eight miles long.

00:10 And at some point, they'll have run all the way into each other and we'll meet up.

00:16 And so one of them is traveling five miles per hour, and the other is traveling seven miles per hour.

00:21 We not want to know how long until they meet.

00:25 So at some point when they meet, they will travel, have covered together the total of eight miles.

00:32 So i'm going to label this person a and this person b.

00:35 And so this person a went their distance traveled plus b's distance is eight miles.

00:44 That's the time we're looking for it.

00:48 Okay.

00:48 So well how do we know how long that is? so i'm going to call t my variable time in hours.

01:00 Okay.

01:00 So if i have person a is running five, miles per hour times t so times some number of hours plus that's going to give me a distance this is going to give me some amount of mile and then if i have person b is running seven miles per hour for that same amount of time from there at some point they're going to meet and that's going to be after the same amount of time because they will both be there um so these will both be in miles five times some amount of time plus seven times some amount of time it's going to add up to eight miles okay, so here we have an equation we can solve...

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