Math Problem Statement

The position of a particle as a function of time is shown in the figure. What is the particle's average velocity between times t = 0 and t = 8 s?

Solution

It looks like you've uploaded an image containing a graph of the position of a particle over time, along with some related questions. Let's break it down and solve them step by step.

  1. Understanding the Graph:

    • The graph represents the position x(t)x(t) of a particle as a function of time tt.
    • The horizontal axis is time tt (in seconds), and the vertical axis is position xx (in meters).
  2. First Question:

    • "What is the particle's average velocity between times t=0st = 0 \, \text{s} and t=8s?t = 8 \, \text{s}?"

The average velocity vavgv_{\text{avg}} is defined as:

vavg=ΔxΔt=x(tfinal)x(tinitial)tfinaltinitialv_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x(t_{\text{final}}) - x(t_{\text{initial}})}{t_{\text{final}} - t_{\text{initial}}}

Let's calculate the change in position between t=0st = 0 \, \text{s} and t=8st = 8 \, \text{s} based on the graph and then compute the average velocity.

  1. Second Question:
    • "What is the average speed of the particle between times t=6st = 6 \, \text{s} and t=7s?t = 7 \, \text{s}?"

The average speed is the total distance traveled (regardless of direction) divided by the total time.

  1. Third Question:
    • "What is the average acceleration of the particle in the time interval between t=1.5st = 1.5 \, \text{s} and t=4s?t = 4 \, \text{s}?"

Average acceleration aavga_{\text{avg}} is defined as:

aavg=ΔvΔta_{\text{avg}} = \frac{\Delta v}{\Delta t}

We will analyze the velocity changes between t=1.5st = 1.5 \, \text{s} and t=4st = 4 \, \text{s}.

I'll extract the values from the graph and perform the calculations next.

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Math Problem Analysis

Mathematical Concepts

Kinematics
Average Velocity
Acceleration

Formulas

Average velocity: v_avg = (x_final - x_initial) / (t_final - t_initial)
Average speed: speed_avg = total distance / total time
Average acceleration: a_avg = (v_final - v_initial) / (t_final - t_initial)

Theorems

Basic Kinematics Theorems

Suitable Grade Level

Grades 9-12