Math Problem Statement

شکل زیر نمودار مکان – زمان متحرک را نشان می دهد که در امتداد محور x حرکت می کند و تنبی متوسط آن در بازه زمانی صفر تا ۴ برابر t=۴s است. در بازه زمانی که متحرک خود را کم می کند چه مسافتی را بر حسب متر حرکت می کند؟

Solution

This problem involves calculating the average velocity over a specific time interval, using the provided position vs. time graph. Let’s analyze it step-by-step:

Problem Details:

  1. The graph shows the position xx (in meters) of an object moving along the xx-axis as a function of time tt (in seconds).
  2. The total displacement from t=0t = 0 to t=4t = 4 is shown.
  3. You are asked to calculate the average speed of the object during this interval.

Key Formula:

The average speed is calculated using: vavg=Total DistanceTime Intervalv_{\text{avg}} = \frac{\text{Total Distance}}{\text{Time Interval}}


Step 1: Analyze the graph

From the graph:

  • The position starts at x=0x = 0 at t=0t = 0.
  • It reaches a maximum of x=8mx = 8 \, \text{m} at t=2st = 2 \, \text{s}.
  • Then it decreases back to x=0mx = 0 \, \text{m} at t=4st = 4 \, \text{s}.

Step 2: Total distance traveled

The total distance is the sum of the segments:

  1. From t=0t = 0 to t=2t = 2, the object moves from x=0x = 0 to x=8x = 8. Distance = 8m8 \, \text{m}.
  2. From t=2t = 2 to t=4t = 4, the object moves from x=8x = 8 back to x=0x = 0. Distance = 8m8 \, \text{m}.

Thus, the total distance traveled is: Total Distance=8+8=16m.\text{Total Distance} = 8 + 8 = 16 \, \text{m}.


Step 3: Time interval

The total time interval is t=4st=0s=4st = 4 \, \text{s} - t = 0 \, \text{s} = 4 \, \text{s}.


Step 4: Average speed calculation

Substitute the values into the formula for average speed: vavg=Total DistanceTime Interval=16m4s=4m/s.v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Time Interval}} = \frac{16 \, \text{m}}{4 \, \text{s}} = 4 \, \text{m/s}.


Step 5: Check the options

The correct answer is 4m/s4 \, \text{m/s}.


Do you have any further questions or want details on the steps?
Here are 5 related questions to expand your understanding:

  1. How is total displacement different from total distance traveled?
  2. What is the difference between average speed and instantaneous speed?
  3. How would the solution change if the object started at x=2mx = 2 \, \text{m} instead of x=0mx = 0 \, \text{m}?
  4. Can the average velocity ever be negative? Why or why not?
  5. How do we calculate the instantaneous velocity from a position-time graph?

Tip:

For position-time graphs, the slope at any point gives the instantaneous velocity, while the overall change in position divided by time gives the average velocity.

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Math Problem Analysis

Mathematical Concepts

Kinematics
Average Velocity
Distance and Time

Formulas

v_avg = Total Distance / Time Interval

Theorems

The average velocity is the total distance traveled divided by the total time interval.

Suitable Grade Level

Grades 9-12