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Chapter 2: Problem 2

A motorist travels \(80 \mathrm{~km}\) at \(100 \mathrm{~km} / \mathrm{h},\) and \(50 \mathrm{~km}\) at \(75 \mathrm{~km} / \mathrm{h}\). What is the average speed for the trip?

Short Answer

Expert verified
The average speed is approximately 88.57 km/h.

Step by step solution

01

Calculate Time for Each Segment

For the first segment, the motorist travels 80 km at 100 km/h. Time for this segment is calculated as: \( t_1 = \frac{80 \text{ km}}{100 \text{ km/h}} = 0.8 \text{ hours} \). For the second segment, traveling 50 km at 75 km/h, the time is: \( t_2 = \frac{50 \text{ km}}{75 \text{ km/h}} \approx 0.6667 \text{ hours} \).
02

Calculate Total Distance and Total Time

The total distance of the trip is the sum of both segments: \( D_{\text{total}} = 80 \text{ km} + 50 \text{ km} = 130 \text{ km} \). The total time is the sum of the time for each segment: \( T_{\text{total}} = 0.8 \text{ hours} + 0.6667 \text{ hours} = 1.4667 \text{ hours} \).
03

Calculate Average Speed

The average speed for the trip is given by the total distance divided by the total time: \( v_{\text{avg}} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{130 \text{ km}}{1.4667 \text{ hours}} \approx 88.57 \text{ km/h} \).

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Key Concepts

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

Distance Traveled
Understanding distance traveled is essential in mechanics, especially when calculating average speed. When we discuss distance traveled, we refer to the length of the path taken during a journey. In mechanics problems, distance is typically measured in units such as kilometers or miles. For instance, if a motorist travels 80 km and then an additional 50 km, the total distance traveled can be calculated by adding these two segments together. This simple addition gives us the total distance as 130 km. It's crucial to note that distance is a scalar quantity, which means it only has magnitude and no direction. When dealing with problems involving distance, ensure you clearly understand each segment traveled and sum them accurately.
Time Calculation
Accurately calculating time is vital for determining average speed and understanding journey dynamics. Time calculation in a journey depends on both the distance traveled and the speed maintained. Time can be calculated using the formula:- Time = Distance / Speed.
For example, if a motorist travels 80 km at a speed of 100 km/h, you would calculate the time for this segment as:- \( t_1 = \frac{80 \text{ km}}{100 \text{ km/h}} = 0.8 \text{ hours} \).
The second segment of 50 km at a speed of 75 km/h would be calculated as:- \( t_2 = \frac{50 \text{ km}}{75 \text{ km/h}} \approx 0.6667 \text{ hours} \).
Add these times together to get the total time for the entire journey. Time, being a scalar quantity, flows in one direction and is crucial in determining how an object moves over time.
Mechanics Problem
Mechanics problems often involve concepts like average speed, which ties together distance and time. Average speed represents how fast an object is moving on average over a period and is calculated by dividing the total distance traveled by the total time taken. The formula for average speed is:- Average Speed = Total Distance / Total Time.
For illustrative purposes, consider the mechanics problem where a motorist covers a total distance of 130 km with a total time of approximately 1.4667 hours. Therefore, the average speed would be:- \( v_{\text{avg}} = \frac{130 \text{ km}}{1.4667 \text{ hours}} \approx 88.57 \text{ km/h} \).
This formula provides a comprehensive overview of the journey without considering the variations of speed during the trip. Understanding this concept helps in solving a wide range of mechanics problems and calculating other related parameters efficiently.

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Most popular questions from this chapter

A toy rocket is launched (from the ground) vertically upward with a constant acceleration of \(30.0 \mathrm{~m} / \mathrm{s}^{2}\). After traveling \(1000 \mathrm{~m}\), its engines stop. When it reaches the very top of its motion, it falls for 0.500 s before a parachute deploys and it descends safely to the ground at the speed it has at that time. (a) What is the maximum altitude reached by the rocket? (b) How long does the rocket take to get to its maximum altitude? (c) How long does the total trip, from launch to ground impact, take?

An object initially at rest experiences an acceleration of \(2.00 \mathrm{~m} / \mathrm{s}^{2}\) on a level surface. Under these conditions, it travels \(6.00 \mathrm{~m}\). Let's designate the first \(3.00 \mathrm{~m}\) as phase 1 with a subscript of 1 for those quantities, and the second \(3.00 \mathrm{~m}\) as phase 2 with a subscript of \(2 .\) (a) The times for traveling each phase should be related by which condition: (3) \(t_{1}>t_{2} ?\) (b) Now calculate the (1) \(t_{1}

A regional airline flight consists of two legs with an intermediate stop. The airplane flies \(400 \mathrm{~km}\) due north from airport A to airport B. From there, it flies \(300 \mathrm{~km}\) due east to its final destination at airport \(C\). (a) What is the plane's displacement from its starting point? (b) If the first leg takes \(45 \mathrm{~min}\) and the second leg \(30 \mathrm{~min}\), what is the average velocity for the trip? (c) What is the average speed for the trip? (d) Why is the average speed not the same as the magnitude for the average velocity?

A sports car can accelerate from 0 to \(60 \mathrm{mi} / \mathrm{h}\) in \(3.9 \mathrm{~s}\). What is the magnitude of the average acceleration of the car in meters per second squared?

An automobile traveling at \(15.0 \mathrm{~km} / \mathrm{h}\) along a straight, level road accelerates to \(65.0 \mathrm{~km} / \mathrm{h}\) in \(6.00 \mathrm{~s}\). What is the magnitude of the auto's average acceleration?
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