Question

Consider three ice skaters: Anna moves in the positive \(x\) -direction without reversing. Bertha moves in the negative \(x\) -direction without reversing. Christine moves in the positive \(x\) -direction and then reverses the direction of her motion. For which of these skaters is the magnitude of the average velocity smaller than the average speed over some time interval?

Short answer

Answer: Christine

Step-by-step solution

Determine the average speed and average velocity for Anna

Since Anna moves in the positive x-direction without reversing, her motion is always in the same direction. This means that her total distance traveled and displacement will have the same magnitude. Therefore, her average speed and average velocity will have the same magnitude.

Determine the average speed and average velocity for Bertha

Similarly, Bertha moves in the negative x-direction without reversing, which means her motion is always in the same direction. Just like Anna, Bertha's total distance traveled and displacement will have the same magnitude, so her average speed and average velocity will also have the same magnitude.

Determine the average speed and average velocity for Christine

Christine moves in the positive x-direction and then reverses the direction of her motion. This means that her displacement will be different from her total distance traveled as she would be moving in opposite directions. Since the total distance traveled will be greater than the magnitude of the displacement, her average speed will be greater than the magnitude of the average velocity.

Conclusion

Therefore, the only ice skater who has the magnitude of her average velocity smaller than her average speed over some time interval is Christine.

Key concepts

average speed

Average speed is a measure of the total distance traveled by an object divided by the total time it takes to travel that distance. It is a scalar quantity, meaning it doesn’t have a direction, only a magnitude.

In the context of the ice skaters, each one covers a different distance over a certain period, and their average speed is calculated by summing up their respective paths:

• Anna's and Bertha's average speeds are straightforward, as they move in single directions without reversing, so their paths equal their total distances.
• Christine, however, fluctuates in direction, making her path longer since she returns part of the way she came. It's key to remember that this means her average speed is calculated over a greater total distance than the other two skaters, given the same direction changes over time.

displacement

Displacement is a vector quantity, which means it has both magnitude and direction. It represents the straight-line distance from an initial position to a final position, effectively capturing an object's overall change in position.

Anna and Bertha

Since Anna and Bertha skate consistently in one direction, their displacement matches their total distance traveled: the endpoints of their motion don't deviate from a straight line path. Thus, their displacement aligns with their paths in terms of magnitude.

Christine's Path

Christine sets off in one direction, reverses back over some previous distance, creating a unique ending position. This change means that while her distance traveled multiplies due to her direction change, her displacement only measures a direct line from start to finish, making it often less than the journey actually covered.

direction of motion

The direction of motion tells us the path along which a person or object is moving. It's a vital part of understanding the difference between displacement and distance traveled.

Consistent Direction for Anna and Bertha

Anna's and Bertha's motions are in straight lines along the x-axis, without direction changes. They don’t reverse, so direction remains consistent—plus or minus—but steady overall.

Changing Direction for Christine

Christine’s pattern includes running forward and reversing direction. Her varying rule breaks consistency, suggesting why her displacement is purely the net effect of her position's alteration, excluding travels against initial motion, thus affecting her velocity but not her speed as dramatically.

distance traveled

Distance traveled accounts for the entire length covered by a moving object, following every turn or pause. It's a scalar, strictly related to the size of the path taken, not where it leads to.

In comparing the skaters, Anna and Bertha's travels correlate directly with both their travelled distance and displacement, since neither turns back on their path.

Christine diverges here; the effort covers more ground than her start-to-end displacement might suggest. This is because she travels back on her path, effectively increasing her actual traveled path compared to her direct end-to-start line, indicative wholly by the reverses made during her ride.