Runner A is initially 2.4 km west of a flagpole and is running with a constant velocity of 8.4 km/h due east. Runner B is initially 2.2 km east of the flagpole and is running with a constant velocity of 7.4 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?
Answer: ____ km from the flagpole due (east, west, south or north)
Xa= -2.4 + 8.4 t
Xb = 2.2 - 7.4 t
(positive direction is east)
Set Xa = Xb and solve for t (in hours). Use that to calculate Xa or Xb at that time.
-2.4 + 8.4t = 2.2 -7.4 t
t = 0.291 hours = 17.5 minutes
Xa = Xb = 0.047 km (east of flagpole)
To find the distance of the two runners from the flagpole when their paths cross, we can use the concept of relative velocity.
First, let's consider the motion of Runner A. Runner A is initially 2.4 km west of the flagpole and is running with a constant velocity of 8.4 km/h due east. We can represent Runner A's position as a function of time using the equation: distance_A = initial_distance_A + (velocity_A * time).
Runner A's position as a function of time can be expressed as: distance_A = 2.4 km + (8.4 km/h * t).
Now, let's consider the motion of Runner B. Runner B is initially 2.2 km east of the flagpole and is running with a constant velocity of 7.4 km/h due west. Similarly, we can represent Runner B's position as a function of time using the equation: distance_B = initial_distance_B + (velocity_B * time).
Runner B's position as a function of time can be expressed as: distance_B = 2.2 km + (7.4 km/h * t).
Now, let's find the time when their paths cross. When their paths cross, the distances covered by each runner will be equal. Therefore, we can equate the distances covered by Runner A and Runner B and solve for time (t):
2.4 km + (8.4 km/h * t) = 2.2 km + (7.4 km/h * t)
0.2 km + (8.4 km/h * t) = (7.4 km/h * t)
0.2 km = (7.4 km/h - 8.4 km/h) * t
0.2 km = (-1 km/h) * t
t = -0.2 km / (-1 km/h)
t = 0.2 hours
Now, we can substitute the value of t = 0.2 hours into either Runner A or Runner B's position equations to find the distance from the flagpole when their paths cross.
Using Runner A's position equation:
distance_A = 2.4 km + (8.4 km/h * 0.2 hours)
distance_A = 2.4 km + 1.68 km
distance_A = 4.08 km (east of the flagpole)
Therefore, when their paths cross, Runner A will be 4.08 km east of the flagpole.
Similarly, the distance of Runner B from the flagpole can be found by substituting t = 0.2 hours into Runner B's position equation.
Using Runner B's position equation:
distance_B = 2.2 km + (7.4 km/h * 0.2 hours)
distance_B = 2.2 km + 1.48 km
distance_B = 3.68 km (west of the flagpole)
Therefore, when their paths cross, Runner B will be 3.68 km west of the flagpole.
To find the total distance between the two runners and the flagpole, we can sum the distances from the flagpole separately for both runners:
Total distance = distance_A + distance_B
Total distance = 4.08 km + 3.68 km
Total distance = 7.76 km
Therefore, when their paths cross, the distance of the two runners from the flagpole will be 7.76 km from the flagpole to the east.