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Chapter 2: Problem 11

You are to drive to an interview in another town, at a distance of \(300 \mathrm{~km}\) on an expressway. The interview is at \(11: 15 \mathrm{~A} . \mathrm{M}\). You plan to drive at \(100 \mathrm{~km} / \mathrm{h}\), so you leave at \(8: 00\) A.M. to allow some extra time. You drive at that speed for the first \(100 \mathrm{~km}\), but then construction work forces you to slow to \(40 \mathrm{~km} / \mathrm{h}\) for \(40 \mathrm{~km}\). What would be the least speed needed for the rest of the trip to arrive in time for the interview?

Short Answer

Expert verified
The minimum speed required for the remaining distance is 128 km/h.

Step by step solution

01

Calculate Time Spent at Different Speeds

First, calculate the time taken for the first 100 km driven at 100 km/h. \[\text{Time} = \frac{100 \text{ km}}{100 \text{ km/h}} = 1 \text{ hour}\] Next, calculate the time taken to cover the 40 km at 40 km/h.\[\text{Time} = \frac{40 \text{ km}}{40 \text{ km/h}} = 1 \text{ hour}\]Add these two times to find the total time spent so far: \[1 + 1 = 2 \text{ hours}\].
02

Determine Remaining Distance and Time

The total distance of the journey is 300 km. You have already covered 100 km + 40 km = 140 km. The remaining distance is:\[300 \text{ km} - 140 \text{ km} = 160 \text{ km}\]You left at 8:00 A.M. and you have spent 2 hours driving so far, meaning it is now 10:00 A.M. You need to be at your interview by 11:15 A.M., which gives you 1 hour and 15 minutes (or 1.25 hours) to complete the remaining distance.
03

Calculate the Required Speed for Remaining Distance

To find the minimum speed needed to cover the remaining 160 km in 1.25 hours, use the formula for speed:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]Substitute the known values:\[\text{Speed} = \frac{160 \text{ km}}{1.25 \text{ hours}} = 128 \text{ km/h}\]Thus, you need to travel at a minimum speed of 128 km/h to reach the interview on time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Speed Calculation
When tackling problems involving speed, time, and distance, it's crucial to understand the fundamental relationship among them. Speed is essentially how fast an object is moving, calculated using the formula:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]This formula allows you to determine how quickly you are traveling over a certain distance within a given time frame.
For example, if you drove 100 km and it took you 1 hour, your speed would be:
  • 100 km/h, as 100 divided by 1 equals 100.
This simple division helps in understanding how much ground is covered per unit of time. Remember, speed can vary during a trip, requiring separate calculations for each segment, as demonstrated in solving the original exercise problem.
Relative Motions
Relative motion refers to the comparison of motion between two or more objects. In the context of traveling, you might often find yourself changing speeds due to various conditions. This was evident in the original exercise, where construction forced a reduction in speed, thus affecting overall travel time.
Understanding how one change in speed impacts the whole journey is vital in real-life situations. For instance:
  • Initially, traveling at 100 km/h for a certain distance was effective.
  • Slowing down to 40 km/h reduces the distance covered in the same time span.
These adjustments are crucial for accurate timing and planning, especially when there are several segments in a journey with different speed limits or conditions.
Distance-Time Relationship
The link between distance and time is key to comprehending motion and planning travels efficiently. Observing the distance-time relationship means keeping an eye on how the total distance is divided over the possible travel time.
In the scenario provided, the total journey was 300 km, completed in various stages. The first 140 km were done in 2 hours, a straightforward distance-time calculation:
  • 100 km in the first hour,
  • 40 km in the second hour due to reduced speed.
Understanding this relationship ensures you plan and execute travel segments wisely. To cover the remaining 160 km in the remaining 1.25 hours required a recalculation of speed. By dividing the leftover distance by the remaining time, you obtain the necessary speed to keep the appointment schedule tight. This places emphasis on the dynamic relationship between how far you travel and how much time you have left.

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Most popular questions from this chapter

At a construction site a pipe wrench struck the ground with a speed of \(24 \mathrm{~m} / \mathrm{s}\). (a) From what height was it inadvertently dropped? (b) How long was it falling? (c) Sketch graphs of \(y, v\), and \(a\) versus \(t\) for the wrench.

A certain elevator cab has a total run of \(190 \mathrm{~m}\) and a maximum speed of \(305 \mathrm{~m} / \mathrm{min}\), and it accelerates from rest and then back to rest at \(1.22 \mathrm{~m} / \mathrm{s}^{2} .\) (a) How far does the cab move while accelerating to full speed from rest? (b) How long does it take to make the nonstop \(190 \mathrm{~m}\) run, starting and ending at rest?

You are arguing over a cell phone while trailing an unmarked police car by \(25 \mathrm{~m} ;\) both your car and the police car are traveling at \(110 \mathrm{~km} / \mathrm{h}\) Your argument diverts your attention from the police car for \(2.0 \mathrm{~s}\) (long enough for you to look at the phone and yell, "I won't do that!"). At the beginning of that \(2.0 \mathrm{~s}\), the police officer begins braking suddenly at \(5.0 \mathrm{~m} / \mathrm{s}^{2} .\) (a) What is the separation between the two cars when your attention finally returns? Suppose that you take another \(0.40 \mathrm{~s}\) to realize your danger and begin braking. (b) If you too brake at \(5.0 \mathrm{~m} / \mathrm{s}^{2}\), what is your speed when you hit the police car?

A rock is dropped (from rest) from the top of a 60 -m-tall building. How far above the ground is the rock \(1.2 \mathrm{~s}\) before it reaches the ground?

A pilot flies horizontally at \(1300 \mathrm{~km} / \mathrm{h}\), at height \(h=35 \mathrm{~m}\) above initially level ground. However, at time \(t=0\), the pilot begins to fly over ground sloping upward at angle \(\theta=4.3^{\circ}\) (Fig. 2-38). If the pilot does not change the airplane's heading, at what time \(t\) does the plane strike the ground?
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