Chapter 2: Problem 4
Starting from a pillar, you run 200 m east (the \(+x\)-direction) at an average speed of 5.0 m/s and then run 280 m west at an average speed of 4.0 m/s to a post. Calculate (a) your average speed from pillar to post and (b) your average velocity from pillar to post.
Short Answer
Expert verified
Average speed: 4.36 m/s; Average velocity: -0.73 m/s west.
Step by step solution
01
Understand the Problem
You need to calculate both the average speed and the average velocity for a running exercise where the direction changes. Average speed is the total distance traveled divided by the total time taken. Average velocity is the total displacement divided by the total time taken.
02
Calculate Total Distance
First, determine the total distance you run. You run 200 m east and then 280 m west. The total distance is the sum of these distances: 200 m + 280 m = 480 m.
03
Calculate Total Displacement
Displacement is the change in position. You start at the pillar, run 200 m east, and finally 280 m west. Your final displacement from the starting point is 280 m west - 200 m east = 80 m west.
04
Calculate Total Time for East Leg
For the first part of your journey (200 m east), use the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \). Therefore, \( \text{time} = \frac{200 \text{ m}}{5.0 \text{ m/s}} = 40 \text{ s} \).
05
Calculate Total Time for West Leg
For the second part of your journey (280 m west), use the same formula. Therefore, \( \text{time} = \frac{280 \text{ m}}{4.0 \text{ m/s}} = 70 \text{ s} \).
06
Calculate Total Time
Add the times from the east and west legs to get the total time: 40 s + 70 s = 110 s.
07
Calculate Average Speed
Average speed is the total distance divided by total time. So, \( \text{average speed} = \frac{480 \text{ m}}{110 \text{ s}} \approx 4.36 \text{ m/s} \).
08
Calculate Average Velocity
Average velocity is the total displacement divided by total time. The displacement is 80 m west (which could be considered as -80 m assuming east is positive). So, \( \text{average velocity} = \frac{-80 \text{ m}}{110 \text{ s}} \approx -0.73 \text{ m/s} \).
09
Interpret the Numerical Results
The average speed is a measure of how fast you were moving regardless of direction, while the negative sign in average velocity indicates movement in the opposite direction to the positive axis (east).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Average Speed
Average speed is a fundamental concept in kinematics and is defined as the total distance traveled divided by the total time taken to travel that distance. It gives you an idea of how fast you are moving overall. It's important to note that average speed does not care about the direction you're moving in.
To calculate average speed in the exercise, you first determine the total distance. In this case, you run 200 meters east and then 280 meters west. This leads to a total distance of 480 meters. Then, you calculate the total time taken, which is 110 seconds. Using the formula for average speed:
To calculate average speed in the exercise, you first determine the total distance. In this case, you run 200 meters east and then 280 meters west. This leads to a total distance of 480 meters. Then, you calculate the total time taken, which is 110 seconds. Using the formula for average speed:
- Average Speed = Total Distance/Total Time
- \( ext{Average Speed} = \frac{480\text{ m}}{110\text{ s}} \approx 4.36 \text{ m/s} \)
Average Velocity
Average velocity is quite a different concept compared to average speed. It takes into account the direction of the motion and is calculated as the total displacement divided by the total time taken. Displacement, unlike distance, is a vector quantity and considers the overall change in position from the starting point.
For this exercise, first determine the displacement. Start at a pillar, run 200 meters east, then 280 meters west, resulting in a final position 80 meters west of the starting point. With a total time of 110 seconds, average velocity is:
For this exercise, first determine the displacement. Start at a pillar, run 200 meters east, then 280 meters west, resulting in a final position 80 meters west of the starting point. With a total time of 110 seconds, average velocity is:
- Average Velocity = Total Displacement/Total Time
- \( ext{Average Velocity} = \frac{-80\text{ m}}{110\text{ s}} \approx -0.73 \text{ m/s} \)
Displacement
Displacement is a key concept when analyzing movement. Unlike distance, displacement refers to the overall change in position, focusing on where you started versus where you end up. It’s a vector quantity, meaning it has both magnitude and direction.
In this exercise, you start at a pillar, move 200 meters east, and then 280 meters west. To find the displacement, consider the difference between these movements. Technically, you travel 80 meters west from your starting position. This reveals that:
In this exercise, you start at a pillar, move 200 meters east, and then 280 meters west. To find the displacement, consider the difference between these movements. Technically, you travel 80 meters west from your starting position. This reveals that:
- Total Displacement = Final Position - Initial Position
Distance
Distance is the total length of the path traveled, without considering the direction. It accounts for every step taken along the journey and adds up the entire route regardless of the path’s shape or direction.
In our exercise, you ran 200 meters east first, then 280 meters west, covering a total distance of:
In our exercise, you ran 200 meters east first, then 280 meters west, covering a total distance of:
- Total Distance = Sum of All Parts