Problem 4: Average Velocity Analysis
Given Data
- At $t = 1$ s, $x = 9$ m.
- At $t = 3$ s, $x = 17$ m.
- The particle moves continuously in the positive x-direction.
- Average velocity in $[1, 3]$ s equals Average velocity in $[0, 6]$ s.
Calculations
1. Calculate Average Velocity in interval [1, 3]:
$$ v_{avg(1,3)} = \frac{x(3) – x(1)}{3 – 1} $$ $$ v_{avg(1,3)} = \frac{17 – 9}{2} = \frac{8}{2} = 4 \text{ m/s} $$2. Analyze Average Velocity in interval [0, 6]:
We are given that $v_{avg(0,6)} = v_{avg(1,3)} = 4$ m/s.
$$ v_{avg(0,6)} = \frac{x(6) – x(0)}{6 – 0} = 4 $$ $$ x(6) – x(0) = 24 \text{ m} $$Evaluating the Options
Option (a): It was at $x = 5$ m at $t = 0$ s.
If $x(0) = 5$, then $x(6)$ would be $5 + 24 = 29$. While this is a possible valid trajectory (e.g., Uniform motion where $x = 4t + 5$), the problem does not state the motion is uniform. The particle could have started at $x=0$, moved rapidly, then slowed down, as long as the average over 6 seconds is 4 m/s.
Conclusion: Not necessarily true.
Option (b): It is moving with a uniform speed.
The problem states “moving continuously”. It does not imply constant velocity. The velocity could vary (e.g., $v(t) = 3t^2$ or any other function) as long as it satisfies the coordinate constraints.
Conclusion: Not necessarily true.
Option (c): Average velocity in the interval [3, 6] s is 4 m/s.
Let’s check the constraint using the weighted average formula for velocities:
$$ 6 \cdot v_{avg(0,6)} = 1 \cdot v_{avg(0,1)} + 2 \cdot v_{avg(1,3)} + 3 \cdot v_{avg(3,6)} $$ $$ 6(4) = (x(1)-x(0)) + 2(4) + 3 \cdot v_{avg(3,6)} $$ $$ 24 = (9 – x(0)) + 8 + 3 \cdot v_{avg(3,6)} $$ $$ 16 = 9 – x(0) + 3 \cdot v_{avg(3,6)} $$ $$ 3 \cdot v_{avg(3,6)} = 7 + x(0) $$Since $x(0)$ is unknown, $v_{avg(3,6)}$ depends on $x(0)$. It is only 4 m/s if $x(0)=5$. Since we determined in Option (a) that $x(0)$ is not fixed, this option is also not necessarily true.
Option (d): Information is insufficient to decide.
Since we have infinite possible functions $x(t)$ that pass through $(1,9)$ and $(3,17)$ while satisfying the average velocity condition over $[0,6]$, we cannot uniquely determine the initial position, the nature of the motion (uniform or not), or the average velocity in other specific intervals.
Correct Option: (d)