Problem 16 - College Physics

Question

When you normally drive the freeway between Sacramento and San Francisco at an average speed of \(105 \mathrm{~km} / \mathrm{h}\) (65 \(\mathrm{mi} / \mathrm{h}\) ), the trip takes \(1.0 \mathrm{~h}\) and \(20 \mathrm{~min}\). On a Friday afternoon, however, heavy traffic slows you down to an average of \(70 \mathrm{~km} / \mathrm{h}\) (43 \(\mathrm{mi} / \mathrm{h}\) ) for the same distance. How much longer does the trip take on Friday than on the other days?

Step-by-Step Solution

Answer

The trip takes 40 minutes longer on Friday.

Step 1: Convert Time to Hours

First, convert the trip time from hours and minutes to only hours. The trip normally takes 1 hour and 20 minutes, which is equivalent to \( 1 + \frac{20}{60} = \frac{80}{60} = \frac{4}{3} \) hours or approximately 1.33 hours.

Step 2: Calculate Normal Distance

Calculate the distance of the trip using the normal speed. The formula for distance is \( \, \text{Distance} = \text{Speed} \times \text{Time} \, \). Using the normal speed of 105 km/h and time of \( \frac{4}{3} \) hours, the distance is \( 105 \text{ km/h} \times \frac{4}{3} \text{ h} = 140 \text{ km} \).

Step 3: Calculate Friday Travel Time

Now, calculate the travel time on Friday using the slower speed of 70 km/h. Use the distance of 140 km: \( \, \text{Time} = \frac{\text{Distance}}{\text{Speed}} \, \). The time in slower traffic is \( \frac{140 \text{ km}}{70 \text{ km/h}} = 2 \text{ hours} \).

Step 4: Compute Time Difference

Find the difference between the slow Friday travel time and the normal travel time. Subtract the normal time from the Friday time: \( 2 \text{ hours} - \frac{4}{3} \text{ hours} = \frac{6}{3} \text{ hours} - \frac{4}{3} \text{ hours} = \frac{2}{3} \text{ hours} \), which is approximately 40 minutes.

Key Concepts

Speed and Distance Calculations

Understanding how to calculate speed and distance is key to solving many physics problems. Essentially, speed defines how fast something moves over a period, while distance tells you how far it moves.
The formula to remember is:

• Distance = Speed × Time

In our problem, when traveling at an average speed of 105 km/h, the distance was calculated as 140 km using:

• 140 km = 105 km/h × 1.33 h

This demonstrates using speed to calculate how far you have traveled during a specific time frame. Knowing either two of these three elements—speed, distance, or time—allows you to calculate the third.

Time Conversion

Time conversion is the process of expressing time in different units to make calculations easier, often needed in physics problems. In our scenario, we converted the trip time of 1 hour and 20 minutes into hours.
To convert minutes to hours, use:

• 1 hour 20 minutes = 1 + \(\frac{20}{60}\) hours = \(\frac{4}{3}\) hours or approximately 1.33 hours

This conversion is crucial for consistency in unit calculations, as most physics equations require consistent units.

Travel Time Analysis

Analyzing travel time allows us to understand how long a journey takes under different conditions. In our problem, you calculate travel time by considering changes in speed.
On a normal day, traveling at 105 km/h, the journey takes 1.33 hours. By slowing down to 70 km/h on a busy Friday, the time of travel increases:
Time for Friday:

• Time = \( \frac{140\text{ km}}{70\text{ km/h}} = 2\text{ hours} \)

This analysis compares travel times to highlight the effect of slower speeds on overall journey duration.

Traffic Impact on Travel

Traffic can significantly alter travel times by reducing the speed at which you travel. In this case, on a Friday afternoon, heavy traffic reduced the speed from 105 km/h to 70 km/h.
As a result, the trip took longer. Here, the Friday travel time increased to 2 hours. In comparison, the same trip without traffic would take only 1.33 hours.
Thus, the impact of traffic was an additional \(\frac{2}{3}\) hour or roughly 40 minutes, showing how crucially traffic affects travel efficiency.

Step-by-Step Physics Solutions

Learning to solve physics problems involves following a step-by-step approach for clarity and accuracy. Breaking down the problem into manageable steps ensures understanding and minimizes errors.
For this exercise:

• First, convert all time to a consistent unit, such as hours.
• Then, calculate distance using speed and time.
• Next, find the new travel time using slower speeds.
• Finally, determine the time difference caused by traffic delay.

Following each step methodically makes even complex problems easier to tackle and understand.