1. **State the problem:** We have an object moving along the x-direction starting from rest and its distance traveled is recorded every second for 6 seconds. We want to analyze the motion using the given data and kinematic equations, then plot graphs for distance vs. time, distance vs. time squared, and acceleration vs. time. 2. **Given data:** | Time (s) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |----------------|---|---|---|---|---|---|---| | Distance (m) | 0 | 3 | 12| 27| 48| 75|108| | Time Squared (s^2)| 0 | 1 | 4 | 9 |16 |25 |36 | | Acceleration (m/s^2)| 0| 0 | 6 | 6 | 6 | 6 | 6 | | Velocity (m/s) | 0 | 6 |12 |18 |24 |30 |36 | 3. **Analyze the motion using kinematics:** - Starts from rest: initial velocity $v_0=0$. - Constant acceleration after $t>1$ second: $a=6$ $m/s^2$. Using the formula for displacement under constant acceleration from rest: $$s = \frac{1}{2}at^2$$ Check for $t\geq 1$: - For example, at $t=2$: $$s = \frac{1}{2} \times 6 \times 2^2 = 12 \text{ m}$$ which matches the table. 4. **Explain graphs:** **a. Distance vs. Time:** - Non-linear curve increasing faster as time progresses due to acceleration. **b. Distance vs. Time Squared:** - A straight line passing through origin since $s \propto t^2$. **c. Acceleration vs. Time:** - Zero at $t=0$ and $t=1$, then constant at 6 $m/s^2$ afterwards. 5. **Desmos graphs** `distance vs time`: $$y = 3t^2/2$$ for $t>1$ simplified from data (using given acceleration 6). `distance vs time squared`: $$y = 3x$$ where $x = t^2$ `acceleration vs time`: $$y = 6$$ constant acceleration after 1 sec.