91Ó°ÊÓ

Relative velocity is a concept that describes the velocity of an object as observed from another object's frame of reference. In this exercise, we consider the velocities of Runner A and Runner B with respect to each other, as well as with respect to the flagpole.

Both runners are moving towards each other from opposite directions. Runner A is headed east, while Runner B is heading west. Therefore, their relative velocities add up because they are approaching each other, not moving away.

• Runner A's velocity is 9.0 km/h east.
• Runner B's velocity is -8.0 km/h east (or 8.0 km/h west).
• The relative velocity between them is thus 9 + 8 = 17 km/h.

Understanding relative velocity helps to predict the rate at which two objects will meet, which is crucial in solving this problem.

Motion equations are essential tools in kinematics, allowing us to describe object motion quantitatively. For Runner A and Runner B, we use these equations to track their movement over time. In this exercise, the motion equations are used to determine when the two runners will cross paths.

For Runner A, moving towards the east and starting 6 km to the west, the equation of motion is:
\[\text{Distance of A} = 6 + 9t\]
For Runner B, moving west and starting 5 km to the east, the equation of motion is:
\[\text{Distance of B} = 5 + 8t\]
Here:

• "6" is the initial distance of Runner A from the flagpole, westward.
• "5" is the initial distance of Runner B from the flagpole, eastward.
• "t" represents the time in hours.
• "9t" and "8t" are the distances traveled after time "t" for Runners A and B respectively.

These equations allow us to calculate exact positions at any time "t", enabling us to find when the runners' positions are identical - meaning their paths cross.

Problem-solving in physics involves breaking down complex situations into simpler, more manageable parts. In our exercise, we have approached this systematically by using equations of motion and concepts of relative velocity. Here's how we solved the problem:

1. **Understanding the Problem**: We noted both runners' starting positions and velocities.2. **Setting Up Equations**: Using motion equations, we defined how far each runner would be from the flagpole over time.3. **Equating Distances**: To find when they cross paths, we needed the distances from the flagpole to be equal, setting the equations of motion equal to one another.
\[ 6 + 9t = 5 + 8t \]

4. **Solving for Time**: By simplifying, we found that they cross after 1 hour.

5. **Determining Cross Point**: Calculating either runner's distance from the flagpole at this time involves subbing the time back into an equation of motion.
The runners cross 15 km from the flagpole. This structured problem-solving approach helps in understanding not just how but why these calculations yield the solution. Always break tasks into smaller parts, solve systematically, and double-check your steps.