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 A
B. column B
C. column C
D. column D
E. column E
Answer (1)
Column A's distances follow d ( t ) = t 2 , showing a quadratic relationship.
Columns B and C have constant first differences, indicating linear relationships.
Columns D and E do not have constant first or second differences, so they are not quadratic.
Thus, column A represents a quadratic relationship: A .
Explanation
Understanding Quadratic Relationships We are given a table of time versus distance data and need to determine which column of distance data exhibits a quadratic relationship with time. A quadratic relationship can be expressed in the form d ( t ) = a t 2 + b t + c , where d ( t ) is the distance at time t , and a , b , and c are constants. To identify a quadratic relationship, we can examine the second differences of the distance data. If the second differences are constant, the relationship is quadratic.
Analyzing Each Column Let's analyze each column:
Column A: The distances are 0, 1, 4, 9, 16, 25, 36. These values correspond to t 2 for t = 0 , 1 , 2 , 3 , 4 , 5 , 6 . The relationship is d ( t ) = t 2 , which is a quadratic function.
Column B: The distances are 2, 4, 6, 8, 10, 12, 14. The differences between consecutive distances are constant (2). This indicates a linear relationship, not a quadratic one.
Column C: The distances are 9, 18, 27, 36, 45, 54, 63. The differences between consecutive distances are constant (9). This indicates a linear relationship, not a quadratic one.
Column D: The distances are 1, 1, 0.5, 0.33, 0.25, 0.2, 0.16. The differences between consecutive distances are not constant. The second differences are also not constant. This is not a quadratic relationship.
Column E: The distances are 1, 1, 0.25, 0.11, 0.06, 0.04, 0.02. The differences between consecutive distances are not constant. The second differences are also not constant. This is not a quadratic relationship.
Identifying the Quadratic Column Based on the analysis, column A exhibits a clear quadratic relationship where d ( t ) = t 2 . Columns B and C show linear relationships, while columns D and E do not represent quadratic relationships.
Final Answer Therefore, the column of distance data that is possible for a quadratic relationship is column A.
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 how high and how far the ball will travel using quadratic equations. This principle is also applied in engineering to design trajectories for rockets and other projectiles, ensuring they reach their intended targets efficiently.
