Runner A is initially 3.4 km west of a flagpole and is running with a constant velocity of 5.2 km/h due east. Runner B is initially 5.6 km east of the flagpole and is running with a constant velocity of 5.0 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?
just working with the x-axis, or number line, the positions at time t are:
p(A) = -3.4 + 5.2t
p(B) = 5.6 - 5.0t
when they meet,
-3.4+5.2t = 5.6-5.0t
t = .88
p(A) = 1.18
so, the two runners are about 1.2 km east of the pole
To determine the distance at which the two runners cross paths, we need to find the time it takes for them to meet.
Let's consider the motion of Runner A first. Runner A is moving with a constant velocity of 5.2 km/h due east. The distance Runner A needs to cover to reach the point where they cross paths with Runner B is the sum of their initial distances. This is 3.4 km west for Runner A and 5.6 km east for Runner B. So, the total distance Runner A needs to cover is (3.4 km + 5.6 km) = 9 km.
Now, let's calculate the time it takes for Runner A to cover this distance. We can use the formula:
time = distance / speed
In this case, the distance is 9 km and the speed is 5.2 km/h. So:
time = 9 km / 5.2 km/h ≈ 1.73 hours
Similarly, Runner B is moving with a constant velocity of 5.0 km/h due west. We can calculate the time it takes for Runner B to reach the point of intersection using the same formula:
time = distance / speed
The distance for Runner B is also 9 km (same as Runner A) because they are moving towards each other. The speed is 5.0 km/h. So:
time = 9 km / 5.0 km/h = 1.8 hours
The next step is to determine where the runners are at these respective times.
For Runner A, their initial position is 3.4 km west of the flagpole, and they are moving east at a constant velocity. So, after 1.73 hours, Runner A would have covered a distance of:
distance = time × speed = 1.73 hours × 5.2 km/h ≈ 8.98 km
For Runner B, their initial position is 5.6 km east of the flagpole, and they are moving west at a constant velocity. So, after 1.8 hours, Runner B would have covered a distance of:
distance = time × speed = 1.8 hours × 5.0 km/h = 9.0 km
Now we can find the distance between the two runners when they cross paths by subtracting their individual distances from the flagpole:
distance of the two runners from the flagpole = distance of Runner A - distance of Runner B
= 8.98 km - 9.0 km
≈ -0.02 km
The negative sign indicates that Runner A is slightly to the west of the point of intersection. The magnitude of the distance is approximately 0.02 km.
Therefore, the distance of the two runners from the flagpole when their paths cross is approximately 0.02 km.