Question

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

Average speed: 4.36 m/s; Average velocity: -0.73 m/s west.

Step-by-step solution

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.

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.

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.

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, \( \text{time} = \frac{200 \text{ m}}{5.0 \text{ m/s}} = 40 \text{ s} \).

Calculate Total Time for West Leg

For the second part of your journey (280 m west), use the same formula. Therefore, \( \text{time} = \frac{280 \text{ m}}{4.0 \text{ m/s}} = 70 \text{ s} \).

Calculate Total Time

Add the times from the east and west legs to get the total time: 40 s + 70 s = 110 s.

Calculate Average Speed

Average speed is the total distance divided by total time. So, \( \text{average speed} = \frac{480 \text{ m}}{110 \text{ s}} \approx 4.36 \text{ m/s} \).

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, \( \text{average velocity} = \frac{-80 \text{ m}}{110 \text{ s}} \approx -0.73 \text{ m/s} \).

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).

Key concepts

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:

• Average Speed = Total Distance/Total Time

Substitute the values:

• \( ext{Average Speed} = \frac{480\text{ m}}{110\text{ s}} \approx 4.36 \text{ m/s} \)

This tells us that, on average, you are moving at a speed of 4.36 meters per second, regardless of your changes in direction.

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:

• Average Velocity = Total Displacement/Total Time

Substituting the values, assuming west is negative:

• \( ext{Average Velocity} = \frac{-80\text{ m}}{110\text{ s}} \approx -0.73 \text{ m/s} \)

The negative sign indicates that the average direction of movement is towards the west. This shows average velocity provides more detailed directional information compared to speed.

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:

• Total Displacement = Final Position - Initial Position

This difference effectively gives 80 meters west, demonstrating displacement as the net change in position, not the distance moved along the path. In physics problems, assigning a direction with positive or negative signs helps in understanding the movement relative to chosen orientation, such as east being positive.

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:

• Total Distance = Sum of All Parts

This sums up to 480 meters. Distance is always a positive value and gives insight into how much ground has been covered, but it doesn't show whether you ended up near your starting point or not. Focusing purely on the distance gives a complete tally of movement without consideration of ending location.