91Ó°ÊÓ

An astronaut has left the Intemational Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a 10 -s interval. What are the magniude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the positive direction is to the right. (a) At the beginning of the interval the astronaut is moving toward the right along the \(x\) -axis at \(15.0 \mathrm{m} / \mathrm{s},\) and at the end of the interval she is moving toward the right at 5.0 \(\mathrm{m} / \mathrm{s}\) . (b) At the beginning she is moving toward the left at \(5.0 \mathrm{m} / \mathrm{s},\) and at the end she is moving toward the left at 15.0 \(\mathrm{m} / \mathrm{s}\) . (c) At the beginning she is moving toward the right at \(15.0 \mathrm{m} / \mathrm{s},\) and at the end she is moving toward the left at 15.0 \(\mathrm{m} / \mathrm{s}\) .

Step by step solution

01

Understanding the Formula for Average Acceleration

The average acceleration \( \bar{a} \) over a time interval \( \Delta t \) is calculated as: \[ \bar{a} = \frac{\Delta v}{\Delta t} \] where \( \Delta v \) is the change in velocity. We'll apply this formula to each scenario, considering a 10-second interval (\( \Delta t = 10 \text{s} \)).

02

Scenario A: Calculate Change in Velocity

In scenario (a), the astronaut's velocity changes from \( 15.0 \ \text{m/s} \) (right) to \( 5.0 \ \text{m/s} \) (right). Thus, \( \Delta v = 5.0 \ \text{m/s} - 15.0 \ \text{m/s} = -10.0 \ \text{m/s} \).

03

Scenario A: Calculate Average Acceleration

Using the formula, the average acceleration \( \bar{a} \) for scenario (a) is: \[ \bar{a} = \frac{-10.0 \ \text{m/s}}{10 \ \text{s}} = -1.0 \ \text{m/s}^2 \] The negative sign indicates acceleration to the left.

04

Scenario B: Calculate Change in Velocity

In scenario (b), the astronaut's velocity changes from \( -5.0 \ \text{m/s} \) (left) to \( -15.0 \ \text{m/s} \) (left). Thus, \( \Delta v = -15.0 \ \text{m/s} - (-5.0 \ \text{m/s}) = -10.0 \ \text{m/s} \).

05

Scenario B: Calculate Average Acceleration

Using the formula, the average acceleration \( \bar{a} \) for scenario (b) is: \[ \bar{a} = \frac{-10.0 \ \text{m/s}}{10 \ \text{s}} = -1.0 \ \text{m/s}^2 \] The negative sign indicates acceleration to the left.

06

Scenario C: Calculate Change in Velocity

In scenario (c), the astronaut's velocity changes from \( 15.0 \ \text{m/s} \) (right) to \( -15.0 \ \text{m/s} \) (left). Thus, \( \Delta v = -15.0 \ \text{m/s} - 15.0 \ \text{m/s} = -30.0 \ \text{m/s} \).

07

Scenario C: Calculate Average Acceleration

Using the formula, the average acceleration \( \bar{a} \) for scenario (c) is: \[ \bar{a} = \frac{-30.0 \ \text{m/s}}{10 \ \text{s}} = -3.0 \ \text{m/s}^2 \] The negative sign indicates acceleration to the left.

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 91Ó°ÊÓ!

Key Concepts

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

Change in Velocity

Change in velocity is a fundamental concept in kinematics, particularly when studying average acceleration. It is calculated as the difference between the final and initial velocities of an object. This change, denoted as \( \Delta v \), can be positive or negative, indicating the direction of acceleration.

• A positive \( \Delta v \) means the object's speed is increasing in the positive direction.
• A negative \( \Delta v \) signifies a decrease in speed or an increase in speed in the negative direction.

In the context of our astronaut’s movement, note that each segment of the exercise specifies the astronaut's velocity at the start and end of a 10-second period. For example, if the astronaut moves from 15.0 m/s to 5.0 m/s to the right, it reflects a decrease in speed. Conversely, a change from 15.0 m/s to -15.0 m/s implies a reversal in motion from right to left, leading to a larger \( \Delta v \). Understanding the directionality of both initial and final velocities is key in calculating accurate acceleration values.

Kinematics

Kinematics focuses on describing motion without considering its causes. It deals with parameters like position, velocity, and acceleration. In our exercise, the focus is on average acceleration, which involves understanding how an astronaut's speed changes over a specific interval.

• Position: Refers to the object's location along an axis (in this exercise, the x-axis).
• Velocity: It is the rate of change of position. Here, initial and final velocities are crucial for calculating changes.
• Acceleration: Describes how quickly the velocity of an object changes.

Kinematics equations help us quantify these state variables. In this scenario, the average acceleration formula \( \bar{a} = \frac{\Delta v}{\Delta t} \) helps us compute the astronaut's acceleration for different intervals, while inspecting both the magnitude and the sign of this acceleration to infer the direction of the astronaut's movement.

Motion Along the X-Axis

Motion along the x-axis simplifies analyses by restricting movement to one dimension. It helps in accurately understanding vector quantities like velocity and acceleration, which have both magnitude and direction.

• Rightward Motion: Defined as positive direction in this exercise.
• Leftward Motion: Defined as negative direction, impacting velocity values and acceleration accordingly.

For instance, if the velocity is described as 15.0 m/s to the right, it can be treated as \( +15.0 \) m/s. However, moving left, even at the same speed, changes the velocity to \( -15.0 \) m/s. Understanding motion along the x-axis allows us to see the impact of velocity changes not just in magnitude but also in direction. It's key to see why a positive or negative average acceleration results, clarifying how these align with real-world movement, such as the astronaut's test maneuvers.