Problem 4 - Fundamentals of Physics

Question

A car moves uphill at \(40 \mathrm{~km} / \mathrm{h}\) and then back downhill at \(60 \mathrm{~km} / \mathrm{h}\). What is the average speed for the round trip?

Step-by-Step Solution

Answer

The average speed for the round trip is 48 km/h.

Step 1: Define Total Distance

Let's assume the distance the car travels uphill and then downhill is the same, denoted by \( d \). This will help us calculate the total distance for the round trip, which is \( 2d \).

Step 2: Calculate Time Uphill

To find the time taken to travel uphill, use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). The uphill time is \( \frac{d}{40} \) hours.

Step 3: Calculate Time Downhill

Similarly, compute the time taken to travel downhill with the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). The downhill time is \( \frac{d}{60} \) hours.

Step 4: Compute Total Time for the Trip

The total time for the round trip is the sum of the uphill and downhill times: \( \frac{d}{40} + \frac{d}{60} \).

Step 5: Find Common Denominator

To sum the two fractions, find a common denominator, which is 120. Convert the times: \( \frac{d}{40} = \frac{3d}{120} \) and \( \frac{d}{60} = \frac{2d}{120} \).

Step 6: Sum the Times

Now, add the two times: \( \frac{3d}{120} + \frac{2d}{120} = \frac{5d}{120} \). Simplifying gives \( \frac{d}{24} \) hours.

Step 7: Calculate Average Speed

The average speed is found using the formula \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \). Thus, \( \text{Average Speed} = \frac{2d}{\frac{d}{24}} = \frac{2d \times 24}{d} = 48 \mathrm{~km/h} \).

Step 8: Conclusion

Thus, the average speed for the round trip is \( 48 \mathrm{~km/h} \).

Key Concepts

kinematics

Kinematics is a branch of physics that describes the motion of objects without considering the forces causing it. It's like the storytelling of movement, where the plot includes elements such as speed, direction, and time. For instance, when solving problems like a car moving up and down a hill, you focus only on aspects like how fast the car moves and for how long. You don't worry about why it's at that speed.

A typical scenario in kinematics involves understanding how speed and time factor into distance moved. In this exercise, the car's journey can be thought of as two chapters: uphill and downhill. The car travels at different speeds during these chapters—40 km/h uphill and 60 km/h downhill. The challenge is combining these two speeds to find an average speed for the entire round trip. Through kinematics, you learn to use formulas to relate these elements in meaningful ways. It's like using a map to understand how these movement stories unfold.

speed calculation

Speed calculation is a fundamental concept in kinematics and involves determining how fast an object is moving. Speed is defined as the distance traveled per unit of time. In mathematical terms, it's calculated using the formula \( ext{Speed} = rac{ ext{Distance}}{ ext{Time}} \).

This formula shows that speed can vary: the farther you travel in less time, the faster you are. In the exercise problem, calculating speed for uphill and downhill separately illustrates how different speeds can affect travel time.

• Uphill speed: The car travels at 40 km/h.
• Downhill speed: The speed increases to 60 km/h.

Average speed over a journey involves calculating total distance and dividing by total time. In this exercise, though the car travels equal distances uphill and downhill, the differing speeds affect how quickly each leg of the trip is completed. By understanding speed calculations, you learn to estimate how fast you are traveling relative to the distance covered.

distance-time relationship

The distance-time relationship is a key concept when analyzing motion in physics. It helps you determine how far an object has traveled over a certain period. This relationship relies heavily on understanding speed, as mentioned earlier, and allows tracking an object's progress over time.

In the exercise, the distance-time relationship is visible when the car travels uphill and downhill. You must find out how much time each part of the journey takes to then pinpoint the average speed. Here are some aspects to consider:

• The same distance \( d \) is traveled uphill and downhill.
• Total time combines both uphill and downhill times, shown as \( rac{d}{40} + rac{d}{60} \).

By understanding how distance and time interrelate, you can solve for overall speed, making it easier to conceptualize real-world travel situations. The relationship shows the balance between how much ground you cover and the time spent moving, an essential principle in physics that clarifies many scenarios we encounter daily.