Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Starting from a pillar, you run 200 m east (the \(+x\)-direction) at an average speed of 5.0 m/s and then run 280 m west at an average speed of 4.0 m/s to a post. Calculate (a) your average speed from pillar to post and (b) your average velocity from pillar to post.

Short Answer

Expert verified
Average speed: 4.36 m/s; Average velocity: -0.73 m/s west.

Step by step solution

01

Understand the Problem

You need to calculate both the average speed and the average velocity for a running exercise where the direction changes. Average speed is the total distance traveled divided by the total time taken. Average velocity is the total displacement divided by the total time taken.
02

Calculate Total Distance

First, determine the total distance you run. You run 200 m east and then 280 m west. The total distance is the sum of these distances: 200 m + 280 m = 480 m.
03

Calculate Total Displacement

Displacement is the change in position. You start at the pillar, run 200 m east, and finally 280 m west. Your final displacement from the starting point is 280 m west - 200 m east = 80 m west.
04

Calculate Total Time for East Leg

For the first part of your journey (200 m east), use the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \). Therefore, \( \text{time} = \frac{200 \text{ m}}{5.0 \text{ m/s}} = 40 \text{ s} \).
05

Calculate Total Time for West Leg

For the second part of your journey (280 m west), use the same formula. Therefore, \( \text{time} = \frac{280 \text{ m}}{4.0 \text{ m/s}} = 70 \text{ s} \).
06

Calculate Total Time

Add the times from the east and west legs to get the total time: 40 s + 70 s = 110 s.
07

Calculate Average Speed

Average speed is the total distance divided by total time. So, \( \text{average speed} = \frac{480 \text{ m}}{110 \text{ s}} \approx 4.36 \text{ m/s} \).
08

Calculate Average Velocity

Average velocity is the total displacement divided by total time. The displacement is 80 m west (which could be considered as -80 m assuming east is positive). So, \( \text{average velocity} = \frac{-80 \text{ m}}{110 \text{ s}} \approx -0.73 \text{ m/s} \).
09

Interpret the Numerical Results

The average speed is a measure of how fast you were moving regardless of direction, while the negative sign in average velocity indicates movement in the opposite direction to the positive axis (east).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Average Speed
Average speed is a fundamental concept in kinematics and is defined as the total distance traveled divided by the total time taken to travel that distance. It gives you an idea of how fast you are moving overall. It's important to note that average speed does not care about the direction you're moving in.

To calculate average speed in the exercise, you first determine the total distance. In this case, you run 200 meters east and then 280 meters west. This leads to a total distance of 480 meters. Then, you calculate the total time taken, which is 110 seconds. Using the formula for average speed:
  • Average Speed = Total Distance/Total Time
Substitute the values:
  • \( ext{Average Speed} = \frac{480\text{ m}}{110\text{ s}} \approx 4.36 \text{ m/s} \)
This tells us that, on average, you are moving at a speed of 4.36 meters per second, regardless of your changes in direction.
Average Velocity
Average velocity is quite a different concept compared to average speed. It takes into account the direction of the motion and is calculated as the total displacement divided by the total time taken. Displacement, unlike distance, is a vector quantity and considers the overall change in position from the starting point.

For this exercise, first determine the displacement. Start at a pillar, run 200 meters east, then 280 meters west, resulting in a final position 80 meters west of the starting point. With a total time of 110 seconds, average velocity is:
  • Average Velocity = Total Displacement/Total Time
Substituting the values, assuming west is negative:
  • \( ext{Average Velocity} = \frac{-80\text{ m}}{110\text{ s}} \approx -0.73 \text{ m/s} \)
The negative sign indicates that the average direction of movement is towards the west. This shows average velocity provides more detailed directional information compared to speed.
Displacement
Displacement is a key concept when analyzing movement. Unlike distance, displacement refers to the overall change in position, focusing on where you started versus where you end up. It’s a vector quantity, meaning it has both magnitude and direction.

In this exercise, you start at a pillar, move 200 meters east, and then 280 meters west. To find the displacement, consider the difference between these movements. Technically, you travel 80 meters west from your starting position. This reveals that:
  • Total Displacement = Final Position - Initial Position
This difference effectively gives 80 meters west, demonstrating displacement as the net change in position, not the distance moved along the path. In physics problems, assigning a direction with positive or negative signs helps in understanding the movement relative to chosen orientation, such as east being positive.
Distance
Distance is the total length of the path traveled, without considering the direction. It accounts for every step taken along the journey and adds up the entire route regardless of the path’s shape or direction.

In our exercise, you ran 200 meters east first, then 280 meters west, covering a total distance of:
  • Total Distance = Sum of All Parts
This sums up to 480 meters. Distance is always a positive value and gives insight into how much ground has been covered, but it doesn't show whether you ended up near your starting point or not. Focusing purely on the distance gives a complete tally of movement without consideration of ending location.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 m/s\(^2\). Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 m/s\(^2\). How far is Powers above the ground when the helicopter crashes into 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?

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?

It has been suggested, and not facetiously, that life might have originated on Mars and been carried to the earth when a meteor hit Mars and blasted pieces of rock (perhaps containing primitive life) free of the Martian surface. Astronomers know that many Martian rocks have come to the earth this way. (For instance, search the Internet for "ALH 84001.") One objection to this idea is that microbes would have had to undergo an enormous lethal acceleration during the impact. Let us investigate how large such an acceleration might be. To escape Mars, rock fragments would have to reach its escape velocity of 5.0 km/s, and that would most likely happen over a distance of about 4.0 m during the meteor impact. (a) What would be the acceleration (in m/s\(^2\) and \(g'\)s) of such a rock fragment, if the acceleration is constant? (b) How long would this acceleration last? (c) In tests, scientists have found that over 40\(\text{%}\) of \(\textit{Bacillus subtilis}\) bacteria survived after an acceleration of 450,000\(g\). In light of your answer to part (a), can we rule out the hypothesis that life might have been blasted from Mars to the earth?

A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the car's motion, the eastward component of the car's velocity is given by \(v_x(t) =\) 0.860 m/s\(^3)t^2\). What is the acceleration of the car when \(v_x =\) 12.0 m/s?

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free