Question

Runner A is initially 4.0 km west of a flagpole and is running with a constant velocity of 4.5 km/h due east. Runner B is initially 5.0 km east of the flagpole and is running with a constant velocity of 5.5 km/h due west. How far are the runners from the flagpole when their paths cross?

Answer 1

Approximately
`0.05\; {\rm km}` to the east of the flagpole.

Step-by-step explanation:

The distance from the flagpole when the two runners meet can be found in the following steps:

• Determine the initial distance between the two runners. as well as the speed at which this distance is changing.
• Determine the amount of time it takes before the two runners meet.
• Determine the place where the two runners meet by considering how far each runner has travelled.

The two runners were initially at a distance of
`s = (4.0\; {\rm km}) - (-5.0\; {\rm km}) = 9.0\; {\rm km}` away from each other.

Because the two runners are running toward each other, the speed of one runner relative to the other would be equal to the sum of their speeds relative to the ground:

`v = (4.5) + (5.5) = 10.0\; \text{km/h}`.

For example, from the perspective of runner A, the other runner (runner B) would be running towards runner A at a speed of
`v = 10.0\; \text{km/h}` from
`s = 9.0\; {\rm km}` away. In other words, the distance between the two runners is getting smaller at a rate of
`10.0\; \text{km/h}`.

To find the time required for the two runners to meet, divide the distance between them by the speed of one runner relative to the other:

`\begin{aligned}t &= (s)/(v) \\ &= \frac{9.0\; {\rm km}}{10.0\; \text{km/h}} = 0.90\; \text{hours}\end{aligned}`.

In other words, the two runners would meet
`0.90` hours after they started.

Denote the position of the flagpole as
`0`, and denote displacements to the east as positive. Since runner B was initially at
`5.0\; {\rm km}` to the East of the flagpole, the position of this runner would initially be
`(+5.0)\; {\rm km}`.

The velocity of runner B is
`(-5.5)\; \text{km/h}`. This quantity is negative since this runner is moving to the west in a direction opposite to the east. To find the position of this runner after
`t = 0.90\; \text{hour}`, add initial position to the product of velocity and time:

`(+5.0) + (-5.5)\, (0.90) = 0.05\; {\rm km}`.

Note that value of the new position is positive. Since displacements to the east are considered positive, this position would be to the east of the flagpole.

Therefore, the two runners would meet
`0.05\; {\rm km}` to the east of the flagpole.