The distance versus time plot for a particular object shows a quadratic relationship. Which column of distance data is possible for this situation?
| Time (s) | A. Distance (m) | B. Distance (m) | C. Distance (m) | D. Distance (m) | E. Distance (m) |
|---|---|---|---|---|---|
| 0 | 0 | 2.00 | 9.00 | ‡ | ‡ |
| 1 | 1.00 | 4.00 | 18.00 | 1.00 | 1.00 |
| 2 | 4.00 | 6.00 | 27.00 | 0.50 | 0.25 |
| 3 | 9.00 | 8.00 | 36.00 | 0.33 | 0.11 |
| 4 | 16.00 | 10.00 | 45.00 | 0.25 | 0.06 |
| 5 | 25.00 | 12.00 | 54.00 | 0.20 | 0.04 |
| 6 | 36.00 | 14.00 | 63.00 | 0.16 | 0.02 |
A. column [tex]$A$[/tex]
B. column B
C. column C
D. column D
E. column E
Answer (2)
Column A exhibits a quadratic relationship where distance equals the square of time: d = t 2 .
Column B shows a linear relationship: d = 2 t + 2 .
Column C also demonstrates a linear relationship: d = 9 t + 9 .
Columns D and E do not fit a quadratic relationship.
Therefore, the correct answer is column A: A .
Explanation
Understanding the Problem We are given a table of time vs distance data for 5 different scenarios (columns A, B, C, D, and E). We need to determine which column of distance data is possible for a quadratic relationship with time. A quadratic relationship can be expressed as d = a t 2 + b t + c , where d is the distance, t is the time, and a , b , and c are constants.
Checking Each Column Let's examine each column to see if it fits a quadratic equation.
Analyzing Column A Column A: The distances are 0, 1, 4, 9, 16, 25, 36. This looks like d = t 2 . When t = 0 , d = 0 . When t = 1 , d = 1 . When t = 2 , d = 4 . When t = 3 , d = 9 . When t = 4 , d = 16 . When t = 5 , d = 25 . When t = 6 , d = 36 . This column fits a quadratic relationship.
Analyzing Column B Column B: The distances are 2, 4, 6, 8, 10, 12, 14. This looks like a linear relationship d = 2 t + 2 . When t = 0 , d = 2 . When t = 1 , d = 4 . When t = 2 , d = 6 . When t = 3 , d = 8 . When t = 4 , d = 10 . When t = 5 , d = 12 . When t = 6 , d = 14 . This column fits a linear relationship, not a quadratic one.
Analyzing Column C Column C: The distances are 9, 18, 27, 36, 45, 54, 63. This looks like a linear relationship d = 9 t + 9 . When t = 0 , d = 9 . When t = 1 , d = 18 . When t = 2 , d = 27 . When t = 3 , d = 36 . When t = 4 , d = 45 . When t = 5 , d = 54 . When t = 6 , d = 63 . This column fits a linear relationship, not a quadratic one.
Analyzing Column D Column D: The distances are undefined, 1, 0.50, 0.33, 0.25, 0.20, 0.16. This does not look like a quadratic relationship.
Analyzing Column E Column E: The distances are undefined, 1, 0.25, 0.11, 0.06, 0.04, 0.02. This does not look like a quadratic relationship.
Conclusion Therefore, column A is the only one that exhibits a quadratic relationship with time.
Examples
Understanding quadratic relationships is crucial in physics, especially when analyzing projectile motion. For instance, if you throw a ball, its height over time follows a quadratic path due to gravity. By knowing the initial velocity and angle, you can predict the ball's trajectory and landing point using quadratic equations. This principle applies to various scenarios, from launching rockets to designing amusement park rides, making the study of quadratic relationships highly practical and relevant.
The only column of distance data that shows a quadratic relationship with time is Column A, which follows the equation d = t 2 . Other columns exhibit linear or undefined relationships. Thus, the correct answer is Column A: oxed{A} .
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