Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Catch-Up Time Students are traveling in two cars to a football game 135 miles away. One car travels at an average speed of 45 miles per hour. The second car starts \(\frac{1}{2}\) hour later and travels at an average speed of 55 miles per hour. How long will it take the second car to catch up to the first car?

Step-by-Step Solution

Answer

It will take the second car \(2.75\) hours to catch up to the first car.

Step 1: Calculate the head start distance

Since the first car started half an hour earlier, we need to calculate the distance it has travelled during this time. This is done by multiplying the speed (45 mph) by the time (0.5 hours), resulting in \(45 \times 0.5 = 22.5\) miles.

Step 2: Calculate the time for the second car to catch up

Next, we calculate the time it takes for the second car to catch up to the first car. Since the second car is traveling at a faster speed (55 mph), it makes up \(55 - 45 = 10\) miles per hour more than the first car. Therefore, to make up the 22.5 miles the first car had as a head start, the second car would need \(\frac{22.5}{10}=2.25\) hours.

Step 3: Sum up total time for the second car

Lastly, you need to remember that second car started half an hour later than the first. So the total time it takes for the second car to catch up to the first is the sum of this half an hour and the time calculated in the previous step: \(0.5 + 2.25 = 2.75\) hours. This is the final answer.

Key Concepts

Average Speed

Average speed is a fundamental concept in relative motion problems. It’s the total distance traveled divided by the time it takes. For example, if a car travels 90 miles over a period of 2 hours, its average speed is 45 miles per hour. This can be calculated using the formula:

• Average Speed = Total Distance / Total Time

In the context of our exercise, two cars travel at different average speeds of 45 mph and 55 mph. Their speeds are pivotal in resolving how one catches up to the other.

It’s important to note that average speed is not necessarily the same as instantaneous speed. Instead, it provides a comprehensive picture of movement over a period of time. Understanding average speed enables us to evaluate journey times with greater clarity.

Time Calculation

Time calculation is crucial in understanding how long a journey or a specific segment of motion takes. We often calculate time using the formula:

• Time = Distance / Speed

In the exercise, we needed to find out how long it takes for the second car to catch up to the first, which requires a precise time calculation step.

The second car is traveling 10 miles per hour faster than the first car. With this additional speed, it closes the 22.5-mile gap initially created by the first car's head start. Calculating this time involves dividing the distance (22.5 miles) by the speed difference (10 miles per hour):

• Time = 22.5 miles / 10 mph = 2.25 hours

This calculation tells us that the second car will catch the first car in 2.25 hours, not considering its delayed start.

Distance Calculation

Distance calculation helps us determine the total ground covered in a trip segment or the head start in a competition. In many cases, it’s solved using:

• Distance = Speed × Time

In the exercise, the initial distance covered by the first car during its half-hour lead is critical to resolving the problem. This distance is:

• Distance = 45 mph × 0.5 hours = 22.5 miles

Knowing this distance is essential, as it sets the target for the second car to catch up.

When dealing with relative motion, distance calculations often need to account for starts or delays in timing, creating differences in travel progress. Mastering these calculations makes problems involving multiple moving entities more approachable.