Problem 13 - Fundamentals of Physics

Question

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.

Step-by-Step Solution

Answer

(a) 68.28 km/h; (b) 68.28 km/h; (c) 68.28 km/h; (d) 0 km/h.

Step 1: Calculate Average Speed from San Antonio to Houston

To find the average speed when traveling half the time at different speeds, use the formula \( v_{avg} = \left( \frac{2v_1v_2}{v_1 + v_2} \right) \). Here, \(v_1 = 55\mathrm{~km/h}\) and \(v_2 = 90\mathrm{~km/h}\).Substitute the values:\[v_{avg} = \frac{2 \times 55 \times 90}{55 + 90} = \frac{9900}{145} = 68.28 \mathrm{~km/h}.\]

Step 2: Calculate Average Speed from Houston to San Antonio

To find the average speed for traveling equal distances at different speeds, use the formula:\[v_{avg} = \frac{2d}{\frac{d}{v_1} + \frac{d}{v_2}} = \frac{2v_1v_2}{v_1 + v_2}.\]As it turns out, this situation also leads to the same average since it's equivalent to calculating for half the time:\[v_{avg} = \frac{2 \times 55 \times 90}{145} = 68.28 \mathrm{~km/h}.\]

Step 3: Calculate Average Speed for Entire Trip

The entire trip consists of each journey segment. Since the total distance traveled is the same in both directions with equal time at different speeds on mixed criteria, use:\[v_{avg} = \frac{2 \times 68.28}{2} = 68.28 \mathrm{~km/h}.\]

Step 4: Calculate Average Velocity for the Entire Trip

The average velocity for a round trip where the start and end points are the same is always zero, as the displacement is zero.Therefore, average velocity \(v = 0 \mathrm{~km/h}.\)

Step 5: Sketch X versus T for Part (a)

Create a sketch of position \(x\) versus time \(t\) for the trip from San Antonio to Houston. Since you're moving in a positive direction throughout, the graph is an upward-sloping line. The slope is represented by the average speed of 68.28 km/h for this part of the journey.The average velocity can be found on this sketch as the overall slope of the line from starting point to final position, or rise over run, as x increases continuously.

Key Concepts

Average Velocity

The concept of average velocity might initially seem similar to average speed, but they're distinct. Average velocity is defined as the total displacement divided by the total time of travel. In cases like a round trip, where you start and end at the same point, the displacement is zero. Hence, the average velocity is:

• Average velocity \(v_{avg} = \frac{\text{Displacement}}{\text{Total Time}}\).
• For a round trip resulting in zero displacement, \(v_{avg} = 0 \text{ km/h}\).

This is because even though you do cover distance, the start and end point are identical, resulting in no overall change in position. For the round trip from San Antonio to Houston and back, the average velocity becomes zero.

Constant Speed

Constant speed refers to the condition where an object covers equal distances in equal intervals of time without any variation. During a journey, maintaining a constant speed can simplify calculations and predictions about travel time.

• In real-world scenarios, factors such as traffic or road conditions can cause fluctuations in speed.
• When calculations are based on constant speed, travel time and average speed are straightforward to predict.
• Activities like cruise control in vehicles aim to maintain constant speed.

However, in our exercise, the speed varies between 55 km/h and 90 km/h for equal time intervals or distances, leading to "average speed" rather than constant speed.

Round Trip

A round trip involves traveling to a destination and returning to the original point. For example, traveling from San Antonio to Houston and back is a classic round trip.

• In our example, the same speeds are used for half the time and half the distance in each direction.
• Round trips often show how average speed calculations differ from simple velocity because the return path must also be considered.
• Since the final position is the starting point, the overall displacement is zero.

This key aspect explains why the average velocity for a round trip is zero, not reflecting the distance covered during the journey.

Displacement

Displacement refers to the change in position from a starting point to a final point. It is a vector quantity, meaning it has both magnitude and direction.

• It is unrelated to the path taken and only considers the initial and final locations.
• In a complete round trip, like traveling from a city back to the same city, the displacement is zero.
• This is why, in the exercise, even though you drove hundreds of kilometers, the displacement calculation results in zero.

Understanding displacement is crucial for calculating average velocity, particularly in scenarios where the beginning and ending locations are the same.