To determine under what conditions the magnitude of average velocity of an object is equal to its average speed, let's review the definitions of these two quantities:
1. Average Velocity: Average velocity is defined as the total displacement divided by the total time taken. Mathematically, it is given by:
\(\[
\text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}}
\]\)
2. Average Speed: Average speed is defined as the total distance traveled divided by the total time taken. Mathematically, it is given by:
\(\[
\text{Average Speed} = \frac{\text{Total Distance Traveled}}{\text{Total Time}}
\]\)
For the magnitude of average velocity to equal the average speed, let's consider the relationships between displacement and distance. Displacement is a vector quantity that considers the direction of motion, whereas distance is a scalar quantity that accounts only for the total length of the path traveled.
### Conditions:
The magnitude of the average velocity will be equal to the average speed if:
1. Straight Line Motion in a Single Direction: The object must move in a straight line without changing direction. In this case, the displacement will be equal to the total distance traveled, as there is no change in direction.
To illustrate:
If an object moves from point \(\( A \)\) to point \(\( B \)\) in a straight path without reversing direction, then:
\(\[
\text{Total Displacement} = \text{Total Distance Traveled}
\]\)
### Example Scenario:
Consider an object moving in a straight line from point \(\( A \)\) to point \(\( B \)\):
- If the object travels a distance of 100 meters to the east in 50 seconds, then:
\(\[
\text{Total Displacement} = 100 \text{ meters (east)}
\]\)
\(\[
\text{Total Distance Traveled} = 100 \text{ meters}
\]\)
Hence, the average velocity and average speed both would be:
\(\[
\frac{\text{100 meters}}{\text{50 seconds}} = 2 \text{ meters/second}
\]\)
Therefore, under this condition (straight line motion in a single direction), the magnitude of the average velocity equals the average speed.
