Calculating Runner Distances at Flagpole with Constant Velocities

[pringless]

Runner A is initially 3 mi west of a flagpole and is running with a constant velocity of 6.4 mi/h due east. Runner B is initially 2.4 mi east of the flagpole and is running with a constant velocity of 6.7 mi/h due west. How far are the runners from the flagpole when their paths cross? Anwser in units of mi.

 

[HallsofIvy]

This should be relatively easy since everything- the runners, the flagpole, and their paths are all on a straight line. No vectors or trigonometry needed!

First, how far apart are the two runners when they start?
The problem tells you that one is 3 mi west and the other 2.4 miles east of the flag pole: draw a picture.

You also know how fast the two runners are running toward each other
Since they are running directly towards each other, their combined speed is 6.4+ 6.7= 13.1 mph. You know the distance between them and you know the speed with which they are "closing". How long do they run until they meet? In other words, what is t when when 13.1 t is equal to the original distance between them.

Since you are asked for the position when they meet, you will know have to find how far either one is from the flagpole at that time.
Since A starts 3 mi from the flagpole and runs at 6.4 mph, his distance from the flagpole at time t is 3- 6.4 t. (What does it mean if putting in the t you got above makes this negative?)

 

[M98Ranger]

To calculate the distance of each runner from the flagpole when their paths cross, we can use the formula Distance = Speed x Time. We know that both runners are moving with constant velocities, so we can use this formula to find the time it takes for them to cross paths.

For Runner A, the distance from the flagpole is 3 mi when they start running and their velocity is 6.4 mi/h due east. This means that it will take Runner A 3/6.4 = 0.46875 hours to reach the flagpole.

For Runner B, the distance from the flagpole is 2.4 mi when they start running and their velocity is 6.7 mi/h due west. This means that it will take Runner B 2.4/6.7 = 0.35821 hours to reach the flagpole.

Since both runners start at different distances from the flagpole and are moving at different velocities, we need to find the time it takes for them to cross paths. This can be done by finding the LCM (Least Common Multiple) of the two times calculated above.

LCM(0.46875, 0.35821) = 1.32813 hours

Now that we have the time it takes for both runners to cross paths, we can calculate the distance from the flagpole for each runner using the formula Distance = Speed x Time.

For Runner A, Distance = 6.4 mi/h x 1.32813 hours = 8.5325 mi from the flagpole.

For Runner B, Distance = 6.7 mi/h x 1.32813 hours = 8.9141 mi from the flagpole.

Therefore, when their paths cross, Runner A is 8.5325 mi and Runner B is 8.9141 mi from the flagpole.