Question

Runner A is initially 6.8 km west of a flagpole and is running with a constant velocity of 5.4 km/h due east. Runner B is initially 4.4 km east of the flagpole and is running with a constant velocity of 3.8 km/h due west. What will be the distance of the two runners from the flagpole when their paths cross?

Answer 1

1. Runner $A$ is initially $6.8 \, \text{km}$ west of a flagpole and is running with a constant velocity of $5.4 \, \text{km/h}$ due east. Runner $B$ is initially $4.4 \, \text{km}$ east of the flagpole and is running with a constant velocity of $3.8 \, \text{km/h}$ due west. What will be the distance of the two runners from the flagpole when their paths cross?

where $v_{Ax} = 5.4$ and $v_{Bx} = -3.8$ are the projections of the velocity of the runners into $X$-axis.

When the paths of the runners cross, their coordinates become equal:

$$x_1 = x_2, \quad x_A + v_{Ax} \cdot t = x_B + v_{Bx} \cdot t, \quad t_0 = \begin{array}{c} x_B - x_A \\ v_{Ax} - v_{Bx} \end{array}$$

That is the time, when the runners meet each other.

The distance of the two runners from the flagpole at this time is

$$d = \left| x_1(t_0) \right|, \quad \boxed{d = \left| x_A + v_{Ax} \cdot \frac{x_B - x_A}{v_{Ax} - v_{Bx}} \right|}.$$

Let check the dimension.

$$[d] = km + \frac{km}{h} \cdot \frac{km}{km} = km.$$

Let evaluate the quantity.

$$d = \left| -6.8 + 5.4 \cdot \frac{4.4 - (-6.8)}{5.4 - (-3.8)} \right| = 0.23 \, (km).$$

Answer: $0.23 \, km$.