You drive on Interstate 10 from San Antonio to Houston, half...

You drive on Interstate 10 from San Antonio to Houston, half the time at $55 \mathrm{~km} / \mathrm{h}$ and the other half at $90 \mathrm{~km} / \mathrm{h}$. On the way back you travel half the distance at $55 \mathrm{~km} / \mathrm{h}$ and the other half at $90 \mathrm{~km} / \mathrm{h}$. What is your average speed (a) from San Antonio to Houston, (b) from Houston back to San Antonio, and (c) for the entire trip? (d) What is your average velocity for the entire trip? (e) Sketch $x$ versus $t$ for (a), assuming the motion is all in the positive $x$ direction. Indicate how the average velocity can be found on the sketch.

You drive on Interstate 10 from San Antonio to Houston, half the time at $55 \mathrm{~km} / \mathrm{h}$ and the other half at $90 \mathrm{~km} / \mathrm{h}$. On the way back you travel half the distance at $55 \mathrm{~km} / \mathrm{h}$ and the other half at $90 \mathrm{~km} / \mathrm{h}$. What is your average speed (a) from San Antonio to Houston, (b) from Houston back to San Antonio, and (c) for the entire trip?
(d) What is your average velocity for the entire trip? (e) Sketch $x$ versus $t$ for (a), assuming the motion is all in the positive $x$ direction. Indicate how the average velocity can be found on the sketch.

Key Concepts

Displacement versus Time Graph

A displacement versus time graph represents an object's motion where the slope at any point indicates the instantaneous velocity. The overall average velocity can be visualized as the slope of the line joining the initial and final points on the graph, which represents the net displacement divided by the total time.

Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. It is a scalar quantity that does not consider the direction of travel. When speeds vary over different segments of a journey, the calculation must account for the time or distance spent at each speed, using different averaging methods depending on whether the segments are equal in time or in distance.

Arithmetic Mean

The arithmetic mean is the simple average of values, used when the intervals over which the speeds are maintained are equal in time. In such cases, the average speed is calculated by summing the speeds and dividing by the number of segments. This method is relevant when evaluating segments where the time spent at each speed is the same.

Harmonic Mean

The harmonic mean is used to compute the average speed when the distances traveled at different speeds are equal. It is calculated by taking the reciprocal of the average of the reciprocals of the speeds. This method accounts for the fact that time spent at each speed can vary when distance segments are equal, thereby providing a more accurate measure of the overall average speed in such cases.

Average Velocity

Average velocity is defined as the net displacement divided by the total time taken, making it a vector quantity that incorporates direction. Regardless of the speeds or times in individual segments, if a round trip results in zero net displacement, the average velocity for the entire journey is zero.

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