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You drive in a straight line at \(20.0 \mathrm{m} / \mathrm{s}\) for \(10.0 \mathrm{minutes}\), then at \(30.0 \mathrm{m} / \mathrm{s}\) for another \(10.0 \mathrm{minutes}\). (a) Is your average speed 25.0 \(\mathrm{m} / \mathrm{s},\) more than \(25.0 \mathrm{m} / \mathrm{s},\) or less than \(25.0 \mathrm{m} / \mathrm{s}\) ? Explain. (b) Verify your answer to part (a) by calculating the average speed.

Short Answer

Expert verified
The average speed is exactly 25.0 m/s.

Step by step solution

01

Convert Time to Seconds

First, convert the travel times from minutes to seconds because speed is given in meters per second. For the first segment, you have \( 10.0 \ minutes \times 60 \ \frac{seconds}{minute} = 600 \ seconds \). The same is true for the second segment.
02

Calculate Distance for First Segment

Using the speed formula distance = speed \( \times \) time, calculate the distance for the first segment.\[ distance_1 = 20.0 \ \frac{m}{s} \times 600 \ s = 12000 \ m \]
03

Calculate Distance for Second Segment

Repeat the process for the second segment.\[ distance_2 = 30.0 \ \frac{m}{s} \times 600 \ s = 18000 \ m \]
04

Calculate Total Distance

Add the distances from the two trips to get the total distance.\[ total\ distance = 12000 \ m + 18000 \ m = 30000 \ m \]
05

Calculate Total Time

Add the times for both trips to get the total time.\[ total \ time = 600 \ s + 600 \ s = 1200 \ s \]
06

Calculate Average Speed

Average speed is calculated by dividing the total distance by the total time.\[ average \ speed = \frac{30000 \ m}{1200 \ s} = 25.0 \ \frac{m}{s} \]
07

Comparison with Given Speed

Since the calculated average speed matches the given speed of 25.0 \( \frac{m}{s} \), the average speed is exactly 25.0 \( \frac{m}{s} \).

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Key Concepts

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

Kinematics
Kinematics, a fundamental part of physics, helps us understand motion through equations and formulas. In the world of kinematics, we study objects both at rest and in motion. We focus on simple, yet crucial aspects like position, velocity, and acceleration.
By analyzing these elements, we can predict the future position of an object or even determine its path of motion. In exercises like the one provided, kinematics allows us to precisely compute distances and speeds over time, combining these elements into a cohesive understanding of motion.
When calculating average speed, as in this exercise, kinematics guides us through a step-by-step breakdown of the journey. We divide the motion into segments, determine distances using speeds and time, and then aggregate these results for a comprehensive picture of the motion. By mastering kinematics, you can solve real-world problems, from simple exercises to complex engineering tasks.
Distance-Time Relationship
The distance-time relationship is essential when analyzing motion. It describes how an object's position changes over time. In our exercise, this relationship involves calculating the distance an object travels at a constant speed.
To find the total distance, we break down each segment of the journey separately. First, convert the given time into a compatible unit, such as seconds. This allows for coherent calculations since the speeds are given in meters per second. Then, apply the formula:
  • Distance = Speed \( \times \) Time
\[ Distance = Speed \times Time \]
Perform this calculation for each segment individually. This method provides a clear understanding of each segment's contribution to the total journey.
Finally, by adding the distances from two or more segments, we can determine the total distance covered. This comprehensive approach captures the journey from start to finish, making it crucial for accurately determining the average speed.
Velocity
Velocity is a vector quantity, representing both the speed and direction of an object's motion. Unlike speed, which is scalar, velocity provides a more complete picture by considering both how fast and in what direction an object moves.
In simple, straight-line scenarios like our exercise, velocity simplifies to speed due to no change in direction. This makes calculating average speed straightforward by focusing on a single dimension of movement. However, understanding that velocity includes direction is vital when analyzing more complex movements that involve turns or changes in direction.
By studying velocity, we enhance our understanding of objects in motion beyond just how fast they're moving. This helps in comprehending the forces acting on an object, predicting future motion, and solving problems more accurately.
In exercises where only speed is considered, recognizing the potential for change in direction or acceleration is equally important. Grasping these nuances in velocity can deepen your ability to analyze motion in real-world settings.

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Most popular questions from this chapter

A model rocket blasts off and moves upward with an acceleration of \(12 \mathrm{m} / \mathrm{s}^{2}\) until it reaches a height of \(26 \mathrm{m},\) at which point its engine shuts off and it continues its flight in free fall. (a) What is the maximum height attained by the rocket? (b) What is the speed of the rocket just before it hits the ground? (c) What is the total duration of the rocket's flight?

A car is traveling due north at \(18.1 \mathrm{m} / \mathrm{s}\). Find the velocity of the car after \(7.50 \mathrm{s}\) if its acceleration is (a) \(1.30 \mathrm{m} / \mathrm{s}^{2}\) due north or \((\mathrm{b}) 1.15 \mathrm{m} / \mathrm{s}^{2}\) due south.

Rubber Ducks A severe storm on January 10,1992 , caused a cargo ship near the Aleutian Islands to spill 29,000 rubber ducks and other bath toys into the ocean. Ten months later hundreds of rubber ducks began to appear along the shoreline near Sitka, Alaska, roughly 1600 miles away. What was the approximate average speed of the ocean current that carried the ducks to shore in (a) \(\mathrm{m} / \mathrm{s}\) and (b) mi/h? (Rubber ducks from the same spill began to appear on the coast of Maine in July 2003 .)

A rocket blasts off and moves straight upward from the launch pad with constant acceleration. After 3.0 s the rocket is at a height of \(77 \mathrm{m}\). (a) What are the magnitude and direction of the rocket's acceleration? (b) What is its speed at this time?

A stalactite on the roof of a cave drips water at a steady rate to a pool \(4.0 \mathrm{m}\) below. As one drop of water hits the pool, a second drop is in the air, and a third is just detaching from the stalactite. (a) What are the position and velocity of the second drop when the first drop hits the pool? (b) How many drops per minute fall into the pool?

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