/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Two runners start simultaneously... [FREE SOLUTION] | 91Ó°ÊÓ

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Two runners start simultaneously at opposite ends of a 200.0 \(\mathrm{m}\) track and run toward each other. Runner \(A\) runs at a steady 8.0 \(\mathrm{m} / \mathrm{s}\) and runner \(B\) runs at a constant 7.0 \(\mathrm{m} / \mathrm{s} .\) When and where will these runners meet?

Short Answer

Expert verified
The runners meet after 13.33 seconds, with Runner A having traveled approximately 106.64 meters.

Step by step solution

01

Define the Given Values

We have two runners, A and B. Runner A's speed is 8.0 m/s and Runner B's speed is 7.0 m/s. They start 200 meters apart on a track and are running towards each other. We need to determine the time at which they will meet and the distance each runner travels until they meet.
02

Calculate Combined Speed

To find the time they will meet, calculate their combined speed. The combined speed is the sum of their individual speeds:\[\text{Combined speed} = 8.0 \text{ m/s} + 7.0 \text{ m/s} = 15.0 \text{ m/s}.\]
03

Find the Time to Meet

Use the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \) to calculate the time it takes for the runners to meet. The distance is 200 meters, and their combined speed is 15 m/s:\[\text{Time} = \frac{200 \text{ m}}{15 \text{ m/s}} = \frac{200}{15} \text{ s} \approx 13.33 \text{ seconds}.\]
04

Calculate Distance Traveled by Runner A

Determine how far Runner A travels in 13.33 seconds by using the formula \( \text{distance} = \text{speed} \times \text{time} \). Runner A's speed is 8.0 m/s:\[\text{Distance}_A = 8.0 \text{ m/s} \times 13.33 \text{ s} = 106.64 \text{ m}.\]
05

Calculate Distance Traveled by Runner B

Similarly, find the distance Runner B travels in 13.33 seconds. Runner B's speed is 7.0 m/s:\[\text{Distance}_B = 7.0 \text{ m/s} \times 13.33 \text{ s} = 93.31 \text{ m}.\]
06

Confirm the Distances Sum to Total Track Length

Verify that the sum of distances traveled by both runners equals the initial distance of 200 meters:\[106.64 \text{ m} + 93.31 \text{ m} = 199.95 \text{ m} \approx 200 \text{ m}.\]The slight discrepancy is due to rounding; thus the calculation is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Relative Velocity
In kinematics, the concept of relative velocity is essential for problems involving two moving objects. It helps us figure out how fast they are approaching or separating from each other. In this runner's problem, "relative velocity" is the summation of the two speeds because the runners move towards each other.
- Relative velocity is calculated by taking the sum of the speeds of two bodies if they are moving towards each other. - Conversely, if they were moving apart, you would subtract one velocity from the other.
Understanding relative velocity is crucial in analyzing motion as it simplifies the problem. By considering combined speeds, you can quickly find the meeting point of two subjects in motion.
Basics of Motion in Kinematics
Motion is a core concept in physics that pertains to the change in position of an object over time. In physics problems like our runner exercise, motion can often be broken down into simple linear terms, which means moving along a straight path at a consistent speed.
Here’s a breakdown of the fundamentals:
  • Speed: This is the rate at which an object covers distance. For instance, Runner A has a speed of 8 m/s.
  • Distance: The overall length of the path covered by the runners. It is the 200 meters gap in this case.
  • Time: Duration over which motion occurs, needed to determine when the runners meet.
Engaging with basic motion helps you grasp how different variables interact, leading to a thorough understanding of the runners' meet challenge.
Enhancing Problem-Solving Skills
Solving physics problems like this exercise is more than just applying formulas; it's an opportunity to develop key problem-solving skills. Start with breaking down a problem by identifying given information and what you need to find.
Here's how to enhance your skills:
  • Identify knowns and unknowns: Clearly sort out what information is given and what needs to be determined.
  • Logical reasoning: Consider the relationships between the data points and devise a plan to tackle the problem.
  • Step-by-step approach: Follow a systematic process, verifying each step before proceeding to ensure accuracy.
  • Check your work: Always review calculations and results to confirm they make sense in the problem's context.
Developing these skills through exercises prepares you for more complex physics challenges.
The Role of Physics Education
Physics education serves as a foundation for understanding the principles that govern our natural world. Through exercises like the runner problem, students are introduced to essential concepts such as motion, relative velocity, and compound speeds.
Here's why this education is important:
  • Conceptual Understanding: It helps students grasp how and why objects move and interact.
  • Critical Thinking: Solving these problems encourages analytical thinking and reasoning capabilities.
  • Real-World Applications: Principles learned apply to everyday phenomena, making the world more comprehensible.
  • Lifelong Learning: Instills curiosity and equips learners with the skills to explore scientific questions independently.
A solid physics education builds the groundwork for future scientific exploration and innovation.

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Most popular questions from this chapter

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