1. **State the problem:** An automobile starts from rest, accelerates at $4\ \mathrm{m/s^2}$ to a speed of $40\ \mathrm{m/s}$, travels at this constant speed for some time, then decelerates at $5\ \mathrm{m/s^2}$ to rest. The total distance traveled is $1000\ \mathrm{m}$. We need to find: a. Distance covered during acceleration. b. Distance traveled at constant speed. c. Total time of travel. 2. **Formulas and rules:** - Distance during acceleration with constant acceleration: $$s = ut + \frac{1}{2}at^2$$ where $u$ is initial velocity, $a$ acceleration, $t$ time. - Final velocity after acceleration: $$v = u + at$$ - Distance during constant speed: $$s = vt$$ - Distance during deceleration: $$v^2 = u^2 + 2as$$ (here $a$ is negative for deceleration) 3. **Calculate acceleration phase:** - Initial velocity $u = 0$ (starting from rest) - Final velocity $v = 40\ \mathrm{m/s}$ - Acceleration $a = 4\ \mathrm{m/s^2}$ - Time to accelerate: $$t_1 = \frac{v - u}{a} = \frac{40 - 0}{4} = 10\ \mathrm{s}$$ - Distance during acceleration: $$s_1 = ut_1 + \frac{1}{2} a t_1^2 = 0 + \frac{1}{2} \times 4 \times 10^2 = 2 \times 100 = 200\ \mathrm{m}$$ 4. **Calculate deceleration phase:** - Final velocity $v = 0$ (comes to rest) - Initial velocity $u = 40\ \mathrm{m/s}$ - Deceleration $a = -5\ \mathrm{m/s^2}$ - Using $$v^2 = u^2 + 2as$$ to find distance during deceleration $s_3$: $$0 = 40^2 + 2 \times (-5) \times s_3$$ $$0 = 1600 - 10 s_3$$ $$10 s_3 = 1600$$ $$s_3 = 160\ \mathrm{m}$$ 5. **Calculate distance at constant speed:** - Total distance $s_{total} = 1000\ \mathrm{m}$ - Distance at constant speed $s_2 = s_{total} - s_1 - s_3 = 1000 - 200 - 160 = 640\ \mathrm{m}$ 6. **Calculate time at constant speed:** - Speed $v = 40\ \mathrm{m/s}$ - Time at constant speed: $$t_2 = \frac{s_2}{v} = \frac{640}{40} = 16\ \mathrm{s}$$ 7. **Calculate total time of travel:** $$t_{total} = t_1 + t_2 + t_3$$ - Time to decelerate $t_3 = \frac{v - 0}{5} = \frac{40}{5} = 8\ \mathrm{s}$ - So, $$t_{total} = 10 + 16 + 8 = 34\ \mathrm{s}$$ **Final answers:** - a. Distance during acceleration: $200\ \mathrm{m}$ - b. Distance at constant speed: $640\ \mathrm{m}$ - c. Total time of travel: $34\ \mathrm{s}$