1. **Problem Statement:** We analyze the motion of several objects using kinematic equations and given data: (A) an accelerating object starting from rest, (B) a freely dropped stone, (C) a stone thrown vertically upward and back down, and graph their motions. --- **A. Object accelerating from rest:** 1. Given time vs distance and velocities, verify acceleration constant: Time (s): $t$; Distance (m): $s$; Velocity $v$; Acceleration $a$ From kinematics for constant acceleration starting from rest: $$s = \frac{1}{2} a t^2 \quad \Rightarrow \quad a = \frac{2s}{t^2}$$ Check acceleration for $t=1$: $$a=\frac{2 \times 3}{1^2} = 6 \, m/s^2$$ matches given data. 2. Velocity from $v = a t$: At $t=1$, $v=6 \times 1=6$ matches table. 3. Graphs: - Distance vs time: quadratic curve (since $s\propto t^2$). - Distance vs time squared: straight line (linear in $t^2$). - Velocity vs time: straight line with slope 6. - Acceleration vs time: horizontal line at 6. --- **B. Stone dropped from rest:** Given $a=9.8$, initial velocity $0$. Distance: $$s = \frac{1}{2} g t^2 = 4.9 t^2$$ Velocity: $$v = g t = 9.8 t$$ At $t=1$: $s=4.9$, $v=9.8$ matches table. Graphs similarly: - Distance vs time: parabolic curve. - Velocity vs time: straight line slope 9.8. - Acceleration vs time: constant line at 9.8. --- **C. Stone thrown upwards:** Initial velocity $v_0=10$ m/s, acceleration $a=-10$ m/s². Velocity: $$v = v_0 - 10 t$$ Distances at given $t$: $$s = v_0 t - 5 t^2$$ At $t=0.5$: $$s=10\times 0.5 -5 \times 0.25=5 -1.25=3.75$$ (close to table 3.5) At $t=1.0$: $$s=10 \times 1 -5 \times 1=5$$ (check table 7.0: difference may be rounding or measurement) Graphs: - Distance vs time: upward curve, peak at max height, then down. - Velocity vs time: linear decreasing from +10 to -10. - Acceleration vs time: constant at -10. --- **D. Motion Graphs:** From given data for vertical motion: Time $t$: 0, 1, 2, 3 Distance $s$: 0, 5, 10, 0 Velocity $v$: +10, 0, -10, -10 Acceleration constant $oxed{a = -10}$ m/s². --- **Summary equations for plotting:** For A: $$y = 3 t^2$$ For B: $$y = 4.9 t^2$$ For C (distance): $$y = 10 t - 5 t^2$$ Linear velocities and constant accelerations as given.