Question

Starting from the front door of a ranch house, you walk 60.0 m due east to a windmill, turn around, and then slowly walk 40.0 m west to a bench, where you sit and watch the sunrise. It takes you 28.0 s to walk from the house to the windmill and then 36.0 s to walk from the windmill to the bench. For the entire trip from the front door to the bench, what are your (a) average velocity and (b) average speed?

Short answer

Average velocity: -0.3125 m/s; Average speed: 1.5625 m/s.

Step-by-step solution

Understand the Problem

We need to calculate both the average velocity and average speed for the trip described. Remember, average velocity is concerned with the net displacement over time, while average speed considers the total distance traveled over time.

Determine Total Displacement and Total Time

Calculate the overall displacement by finding the difference between the ending and starting positions. The starting position is the front door of the house; the ending position is the bench to the west of the starting point. Total displacement is \(-60 \, \text{m} + 40 \, \text{m} = -20 \, \text{m}\).Total time is the sum of the time for each leg of the journey,28.0 s + 36.0 s = 64.0 s.

Calculate Average Velocity

Average velocity is given by the formula: \(\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}\).Substitute the known values: \(\text{Average velocity} = \frac{-20 \, \text{m}}{64.0 \, \text{s}} = -0.3125 \, \text{m/s}\).

Calculate Total Distance Traveled

The total distance traveled is the sum of both distances: 60.0 m east to the windmill plus 40.0 m west to the bench, which equals 100.0 m.

Calculate Average Speed

Average speed is given by the formula:\(\text{Average speed} = \frac{\text{Total distance traveled}}{\text{Total time}}\).Substitute the known values: \(\text{Average speed} = \frac{100.0 \, \text{m}}{64.0 \, \text{s}} = 1.5625 \, \text{m/s}\).

Key concepts

Average Velocity

Average velocity is an important concept in kinematics. It tells you how fast you are moving in a specific direction over a period of time. It's all about displacement, which is the change in position from the start to the end of a journey.
To calculate average velocity, use the formula:

• \( \text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} \)

In this exercise, you start at a ranch house and end at a bench, resulting in a total displacement of -20 meters. Even though you walked 100 meters in total, the displacement is only 20 meters west (negative means westward here), because east and west cancel each other out.
Divide this displacement by the total time of 64 seconds, and you find an average velocity of -0.3125 meters per second. The negative value indicates the westward direction of the net movement.

Average Speed

Average speed, unlike average velocity, focuses on the total ground covered without considering direction. It provides the overall pace of your movement regardless of where you ended up.Calculate average speed by applying:

• \( \text{Average speed} = \frac{\text{Total distance traveled}}{\text{Total time}} \)

Here, you walked 60 meters east and then 40 meters back west. So, the total distance is a straightforward sum of these two paths, equaling 100 meters.
Dividing 100 meters by the total time of 64 seconds gives you an average speed of 1.5625 meters per second. Unlike average velocity, average speed is always a positive number because it does not account for direction.

Displacement

Displacement is a vector quantity, which means it has both magnitude and direction. It tells you how far out of place an object is; it's the object's overall change in position.In our exercise, displacement is the net result of moving 60 meters east and then 40 meters west. Instead of worrying about the pathway taken, you only consider where you started and where you ended up.

• Calculation: \(-60 \, \text{m} + 40 \, \text{m} = -20 \, \text{m}\)

This means you ended up 20 meters west of your starting point. The negative sign indicates that the ending point is west of the starting point.

Total Distance Traveled

Total distance traveled is a scalar quantity, focusing simply on how much ground was covered during the entire journey. It's about the actual path taken without worrying about direction. To find this, you count every meter walked during the trip:

• You walked 60 meters east to reach the windmill.
• Then, you walked another 40 meters west to get back to the bench.

Add these two distances together to get a total of 100 meters traveled. This total respects every step you took, embodying the complete journey, not just where you ended up relative to the start.