Let's analyze the problem to find the values for [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex] that best complete the chart of the object's velocity over time considering constant acceleration.
We're given four options for [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex]:
1. [tex]\(X = 0\)[/tex], [tex]\(Y = 0\)[/tex], [tex]\(Z = 1\)[/tex]
2. [tex]\(X = 2\)[/tex], [tex]\(Y = 4\)[/tex], [tex]\(Z = 6\)[/tex]
3. [tex]\(X = 3\)[/tex], [tex]\(Y = 3\)[/tex], [tex]\(Z = 3\)[/tex]
4. [tex]\(X = 1\)[/tex], [tex]\(Y = 5\)[/tex], [tex]\(Z = 8\)[/tex]
Since the object is moving with constant acceleration, its velocity should follow a linear pattern, where the change in velocity is the same over equal intervals of time.
Let's evaluate each option:
1. [tex]\(X = 0\)[/tex], [tex]\(Y = 0\)[/tex], [tex]\(Z = 1\)[/tex]
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 1\)[/tex]: Velocity changes from 0 to 0.
- From [tex]\(t = 1\)[/tex] to [tex]\(t = 2\)[/tex]: Velocity changes from 0 to 0.
- From [tex]\(t = 2\)[/tex] to [tex]\(t = 3\)[/tex]: Velocity changes from 0 to 1.
This does not indicate a constant acceleration.
2. [tex]\(X = 2\)[/tex], [tex]\(Y = 4\)[/tex], [tex]\(Z = 6\)[/tex]
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 1\)[/tex]: Velocity changes from 0 to 2.
- From [tex]\(t = 1\)[/tex] to [tex]\(t = 2\)[/tex]: Velocity changes from 2 to 4.
- From [tex]\(t = 2\)[/tex] to [tex]\(t = 3\)[/tex]: Velocity changes from 4 to 6.
This shows a constant acceleration with equal changes (+2) in velocity over each time interval.
3. [tex]\(X = 3\)[/tex], [tex]\(Y = 3\)[/tex], [tex]\(Z = 3\)[/tex]
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 1\)[/tex]: Velocity changes from 0 to 3.
- From [tex]\(t = 1\)[/tex] to [tex]\(t = 2\)[/tex]: Velocity remains 3.
- From [tex]\(t = 2\)[/tex] to [tex]\(t = 3\)[/tex]: Velocity remains 3.
This does not show a constant acceleration; rather, it shows constant velocity from [tex]\(t = 1\)[/tex] onwards.
4. [tex]\(X = 1\)[/tex], [tex]\(Y = 5\)[/tex], [tex]\(Z = 8\)[/tex]
- From [tex]\(t = 0\)[/tex] to [tex]\(t = 1\)[/tex]: Velocity changes from 0 to 1.
- From [tex]\(t = 1\)[/tex] to [tex]\(t = 2\)[/tex]: Velocity changes from 1 to 5.
- From [tex]\(t = 2\)[/tex] to [tex]\(t = 3\)[/tex]: Velocity changes from 5 to 8.
This shows a non-linear increase in velocity, indicating inconsistent acceleration.
The values that best complete the chart for an object moving at a constant acceleration are:
[tex]\[X = 2, Y = 4, Z = 6\][/tex]
