Base Your Answers To Questions 1a And 1c On The Speed-Time G

1base Your Answers To Questions 1a 1c On The Speed Time Graph Belo

1. Base your answers to questions 1a – 1c on the speed-time graph below, which represents the linear motion of a cart. (10 pts) 1a. Determine the magnitude of the acceleration of the cart during interval AB. [Show all calculations, including the equation and substitution with units] 1b. Calculate the distance traveled by the cart during interval BC. [Show all calculations, including the equation and substitution with units] 1c. What is the average speed of the cart during interval CD?

Paper For Above instruction

This analysis examines a linear motion scenario depicted in a speed-time graph, focusing on calculating the acceleration, distance traveled, and average speed during specified intervals. Understanding these concepts provides foundational insights into kinematic principles fundamental to physics and engineering disciplines.

Introduction

Motion analysis through graphs is a vital component of physics education, especially in understanding kinematic concepts such as acceleration, velocity, and displacement. The speed-time graph offers a visual representation of how an object’s speed changes over time, enabling precise quantification of these kinematic quantities. This paper addresses three specific questions based on a provided speed-time graph: the magnitude of acceleration during a given interval, the total distance traveled during another, and the average velocity during a third interval. Accurate interpretation of such graphs requires applying relevant equations from classical mechanics, particularly those relating to uniformly accelerated motion and average speed calculations.

Analyzing the Graph

Given that the graph depicts linear motion, the assumption is that the segments are straight lines, indicative of constant acceleration or uniform motion. Each interval—AB, BC, and CD—represents a specific segment of the motion with distinct kinematic characteristics. To proceed, it is essential to identify the relevant data points (speeds at segment endpoints and time intervals) and apply the appropriate formulas.

1a. Determining the Magnitude of Acceleration during Interval AB

Interval AB represents a period where the speed of the cart changes linearly over time, indicating constant acceleration. The magnitude of acceleration (a) is given by the change in velocity divided by the change in time:

a = Δv / Δt

From the graph, suppose the initial speed at point A is vA and the final speed at point B is vB. The time interval between A and B is ΔtAB. For example, if vA = 2 m/s, vB = 6 m/s, and ΔtAB = 4 s, then:

a = (vB - vA) / ΔtAB = (6 \text{ m/s} - 2 \text{ m/s}) / 4 \text{ s} = 4 \text{ m/s} / 4 \text{ s} = 1 \text{ m/s}^2

Therefore, the magnitude of acceleration during interval AB is 1 m/s². Precise calculation depends on the actual data from the graph; assuming representative values aids illustration.

1b. Calculating the Distance Traveled During Interval BC

During interval BC, the motion may be uniformly accelerated or at constant speed, depending on the graph. If the speed varies linearly, the area under the speed-time curve corresponds to the displacement (distance traveled). The area of the trapezoid formed by the speed at points B and C over the time interval gives the distance:

Distance = Area under the curve = [(vB + vC) / 2] × ΔtBC

Assuming vB = 6 m/s, vC = 4 m/s, and ΔtBC = 3 s, then:

Distance = [(6 \text{ m/s} + 4 \text{ m/s}) / 2] × 3 \text{ s} = (10 \text{ m/s} / 2) × 3 \text{ s} = 5 \text{ m/s} × 3 \text{ s} = 15 \text{ meters}



1c. Computing the Average Speed During Interval CD



The average speed over an interval is the total displacement divided by the time taken or equivalently, the average of the initial and final speeds if the acceleration is uniform. Using the latter method:



Average speed = (vC + vD) / 2





Suppose vC = 4 m/s and vD = 2 m/s. If the duration of interval CD (ΔtCD) is 5 seconds, then the average speed is:



Average speed = (4 \text{ m/s} + 2 \text{ m/s}) / 2 = 3 \text{ m/s}





This value indicates that in the interval CD, the cart's velocity averages at 3 m/s, reflecting deceleration or varying speed as depicted in the graph.



Discussion and Conclusion



Analyzing motion through graphs allows for precise calculations essential in kinematic studies. The key to accurate interpretation lies in correctly identifying the values from the graph—speeds and time intervals—and applying fundamental equations. The calculations illustrate the importance of understanding how displacement correlates with the area under the speed-time curve, how acceleration relates to the change in velocity over time, and how average speed provides a meaningful measure of motion over specified intervals.




These calculations are integral to various applications, including vehicle dynamics, robotics, and physics education. Further detailed analysis requires specific graph data; however, the principles demonstrated here provide a robust framework for interpreting similar kinematic situations.



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