Question

Triple Choice Suppose you ride a bicycle around the block, returning to your starting point. At the end of your trip, is your average speed greater than, less than, or equal to the magnitude of your average velocity? Explain.

Short answer

Average speed is greater than the magnitude of average velocity.

Step-by-step solution

Understanding Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. Since you rode around the block and returned to your starting point, you completed a loop. Let's say the block is a square or circular track, the total distance you traveled is equal to the perimeter of the block.

Understanding Average Velocity

Average velocity is defined as the total displacement divided by the total time taken. Displacement is a vector quantity that refers to the change in position from the start to the end. In this case, because you returned to your starting point, your displacement is zero.

Calculating Average Speed

Suppose the perimeter of the block is \(D\) and the time taken to complete the loop is \(T\). The average speed is calculated as \( \text{Average Speed} = \frac{D}{T} \). This is a positive value since distance and time are both positive.

Calculating Average Velocity

Given that you returned to your starting point, the displacement is zero. Therefore, the average velocity is \( \text{Average Velocity} = \frac{0}{T} = 0 \).

Comparing Average Speed and Average Velocity

Now compare the average speed and the magnitude of the average velocity. The average speed is a positive value, whereas the magnitude of the average velocity is zero. Therefore, the average speed is greater than the magnitude of the average velocity.

Key concepts

Displacement

When considering the concept of displacement, it's important to understand that it is a measure of change in position from the starting point to the ending point. Displacement is not concerned with the path taken, but rather the shortest straight line distance from beginning to end. This path is described by a vector, having both a magnitude and a direction, which sets it apart from other measurement forms that only track the distance.
In your bicycle ride scenario, even if you pedal vigorously around the block, when you return to your starting position, your displacement is zero. This is because you end up at the same point from where you started, essentially having no net change in position. Thus, mathematically, you can express displacement as zero in such a closed-loop journey, regardless of the actual distance traveled.

Vector Quantity

In physics, a vector quantity is one that is characterized by both magnitude and direction. This is in contrast to scalar quantities, which are described solely by magnitude.
Velocity is a prime example of a vector quantity as it indicates not only how fast something is moving, or its speed, but also in which direction. When you're riding your bicycle, your velocity changes as you navigate different parts of the block. Its direction can point north, south, east, or west at various points in your journey, leading up to your initial starting point.
A helpful way to visualize vector quantities is as arrows pointing in the direction of travel, with their length proportional to the speed. Because vectors account for direction, ending back where you started nullifies your displacement vector, making it zero even if considerable distance or speed occurred during the trip.

Speed Calculation

Speed is calculated by dividing the total path or distance traveled by the time taken, making it a scalar quantity. It only concerns how fast something is moving, absent of any direction, which differentiates it from velocity.
In the bicycle exercise, if you know that the total perimeter of the block is represented as \( D \), and the total time taken to ride around it is \( T \), then you can determine the average speed with the formula:

• \( \text{Average Speed} = \frac{D}{T} \)

This calculation results in a positive value because both distance (\( D \)) and time (\( T \)) are always positive.
Hence, while your average speed accounts for the actual distance your wheels have spun, reflecting your physical effort, it doesn’t reflect your displacement. The average speed remains positive and quantifiable even when the average velocity, which considers direction and position change, is zero, further distinguishing the concepts of speed and velocity.