Carlo's journey

Problem

At time $t = 4\;\text{s}$ Carlo is at km 2 and is travelling at $6\;\text{m/s}$. He accelerates for $4\;\text{s}$ with $a = 3\;\text{m/s}^2$, then continues at constant velocity for $100\;\text{m}$, and finally brakes to a stop over $500\;\text{m}$. Determine all the events.

Solution

Setting up. We choose the axis $s$ oriented in the direction of motion, with the origin of times and positions fixed by the initial data. The motion consists of 3 stages and 4 events (A, B, C, D):

• Stage A$\to$B: uniformly accelerated motion ($a = 3\;\text{m/s}^2$) for $4\;\text{s}$;
• Stage B$\to$C: uniform motion ($a = 0$) for $100\;\text{m}$;
• Stage C$\to$D: braking to a stop over $500\;\text{m}$.

It is convenient to organise the data in an events table, filling it in stage by stage:

Event	$t$ (s)	$s$ (m)	$v$ (m/s)	$a$ (m/s²)
A	$4$	$2000$	$6$
				$3$
B	$8$	$2048$	$18$
				$0$
C	$13{,}56$	$2148$	$18$
				$-0{,}324$
D	$69{,}1$	$2648$	$0$

Stage A$\to$B (acceleration). The final velocity and the final position are:

$\begin{aligned} v_B &= v_A + a\,(t_B - t_A) = 6 + 3\cdot 4 = 18\;\text{m/s} \\ s_B &= s_A + v_A\cdot 4 + \tfrac{1}{2}\cdot 3\cdot 4^2 = 2000 + 24 + 24 = 2048\;\text{m} \end{aligned}$

Stage B$\to$C (constant velocity, $a = 0$). The time to cover the $100\;\text{m}$ is:

$t_C = t_B + \frac{\Delta s}{v_B} = 8 + \frac{100}{18} \approx 13{,}56\;\text{s} \qquad s_C = 2048 + 100 = 2148\;\text{m}$

Stage C$\to$D (braking, $v_D = 0$, $\Delta s = 500\;\text{m}$). We obtain the acceleration from the time-independent relation:

$v_D^2 - v_C^2 = 2a\,\Delta s \quad\Rightarrow\quad 0 - 18^2 = 2a\cdot 500 \quad\Rightarrow\quad a = -\frac{324}{1000} = -0{,}324\;\text{m/s}^2$

Finally, the stopping time and the final position:

$t_D = t_C + \frac{v_D - v_C}{a} = 13{,}56 + \frac{0 - 18}{-0{,}324} \approx 69{,}1\;\text{s} \qquad s_D = 2148 + 500 = 2648\;\text{m}$

$\ev{t_D \approx 69{,}1\;\mathrm{s}, \quad s_D = 2648\;\mathrm{m}, \quad v_D = 0}$

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Topics: Cinematica Concepts: Moto uniformemente accelerato Skills: Scelta degli assi · Lettura dei grafici