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Starting from the front door of a ranch house, you walk 60.0 m due east to a windmill, turn around, and then slowly walk 40.0 m west to a bench, where you sit and watch the sunrise. It takes you 28.0 s to walk from the house to the windmill and then 36.0 s to walk from the windmill to the bench. For the entire trip from the front door to the bench, what are your (a) average velocity and (b) average speed?

Short Answer

Expert verified
Average velocity: -0.3125 m/s; Average speed: 1.5625 m/s.

Step by step solution

01

Understand the Problem

We need to calculate both the average velocity and average speed for the trip described. Remember, average velocity is concerned with the net displacement over time, while average speed considers the total distance traveled over time.
02

Determine Total Displacement and Total Time

Calculate the overall displacement by finding the difference between the ending and starting positions. The starting position is the front door of the house; the ending position is the bench to the west of the starting point. Total displacement is \(-60 \, \text{m} + 40 \, \text{m} = -20 \, \text{m}\).Total time is the sum of the time for each leg of the journey,28.0 s + 36.0 s = 64.0 s.
03

Calculate Average Velocity

Average velocity is given by the formula: \(\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}\).Substitute the known values: \(\text{Average velocity} = \frac{-20 \, \text{m}}{64.0 \, \text{s}} = -0.3125 \, \text{m/s}\).
04

Calculate Total Distance Traveled

The total distance traveled is the sum of both distances: 60.0 m east to the windmill plus 40.0 m west to the bench, which equals 100.0 m.
05

Calculate Average Speed

Average speed is given by the formula:\(\text{Average speed} = \frac{\text{Total distance traveled}}{\text{Total time}}\).Substitute the known values: \(\text{Average speed} = \frac{100.0 \, \text{m}}{64.0 \, \text{s}} = 1.5625 \, \text{m/s}\).

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Key Concepts

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

Average Velocity
Average velocity is an important concept in kinematics. It tells you how fast you are moving in a specific direction over a period of time. It's all about displacement, which is the change in position from the start to the end of a journey.
To calculate average velocity, use the formula:
  • \( \text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} \)
In this exercise, you start at a ranch house and end at a bench, resulting in a total displacement of -20 meters. Even though you walked 100 meters in total, the displacement is only 20 meters west (negative means westward here), because east and west cancel each other out.
Divide this displacement by the total time of 64 seconds, and you find an average velocity of -0.3125 meters per second. The negative value indicates the westward direction of the net movement.
Average Speed
Average speed, unlike average velocity, focuses on the total ground covered without considering direction. It provides the overall pace of your movement regardless of where you ended up.Calculate average speed by applying:
  • \( \text{Average speed} = \frac{\text{Total distance traveled}}{\text{Total time}} \)
Here, you walked 60 meters east and then 40 meters back west. So, the total distance is a straightforward sum of these two paths, equaling 100 meters.
Dividing 100 meters by the total time of 64 seconds gives you an average speed of 1.5625 meters per second. Unlike average velocity, average speed is always a positive number because it does not account for direction.
Displacement
Displacement is a vector quantity, which means it has both magnitude and direction. It tells you how far out of place an object is; it's the object's overall change in position.In our exercise, displacement is the net result of moving 60 meters east and then 40 meters west. Instead of worrying about the pathway taken, you only consider where you started and where you ended up.
  • Calculation: \(-60 \, \text{m} + 40 \, \text{m} = -20 \, \text{m}\)
This means you ended up 20 meters west of your starting point. The negative sign indicates that the ending point is west of the starting point.
Total Distance Traveled
Total distance traveled is a scalar quantity, focusing simply on how much ground was covered during the entire journey. It's about the actual path taken without worrying about direction. To find this, you count every meter walked during the trip:
  • You walked 60 meters east to reach the windmill.
  • Then, you walked another 40 meters west to get back to the bench.
Add these two distances together to get a total of 100 meters traveled. This total respects every step you took, embodying the complete journey, not just where you ended up relative to the start.

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Most popular questions from this chapter

During your summer internship for an aerospace company, you are asked to design a small research rocket. The rocket is to be launched from rest from the earth's surface and is to reach a maximum height of 960 m above the earth's surface. The rocket's engines give the rocket an upward acceleration of 16.0 m/s\(^2\) during the time \(T\) that they fire. After the engines shut off, the rocket is in free fall. Ignore air resistance. What must be the value of \(T\) in order for the rocket to reach the required altitude?

A jet fighter pilot wishes to accelerate from rest at a constant acceleration of 5\(g\) to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.0 s. Use 331 m/s for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of 5\(g\) before he blacks out?

A rocket carrying a satellite is accelerating straight up from the earth's surface. At 1.15 s after liftoff, the rocket clears the top of its launch platform, 63 m above the ground. After an additional 4.75 s, it is 1.00 km above the ground. Calculate the magnitude of the average velocity of the rocket for (a) the 4.75-s part of its flight and (b) the first 5.90 s of its flight.

Sam heaves a 16-lb shot straight up, giving it a constant upward acceleration from rest of 35.0 m/s\(^2\) for 64.0 cm. He releases it 2.20 m above the ground. Ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above the ground does it go? (c) How much time does he have to get out of its way before it returns to the height of the top of his head, 1.83 m above the ground?

The rocket-driven sled \(\textit{Sonic Wind No. 2,}\) used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 m (3500 ft) long. Starting from rest, it can reach a speed of 224 m/s(500 mi/h) in 0.900 s. (a) Compute the acceleration in m/s\(^2\), assuming that it is constant. (b) What is the ratio of this acceleration to that of a freely falling body (\(g\))? (c) What distance is covered in 0.900 s? (d) A magazine article states that at the end of a certain run, the speed of the sled decreased from 283 m/s (632 mi/h) to zero in 1.40 s and that during this time the magnitude of the acceleration was greater than 40\(g\). Are these figures consistent?

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