Question

3- mouvement d’un ascenseur /2,5pt
le graphique ci - dessous représente l’évolution de la
coordonné ( v_y ) de l’ascenseur en fonction du temps. l’axe oy nest vertical ascendant. un passager de masse 70kg se trouve
dans la cabine. prendre ( g = 9,8 , \text{m/s}^2 ).
3 - 1- déterminer l’intensité de la réaction ( vec{r} ) du planche sur le
passager au cours de chaque phase du mouvement. 1,5pt
3 - 2- déterminer la distance totale parcourue par l’ascenseur
au cours de son mouvement. 1pt

Explanation:

Step1: Analyze velocity-time graph phases

The graph has 4 phases: 0-10s (acceleration up), 10-30s (constant velocity), 30-40s (deceleration up), 40-50s (free fall? No, deceleration to stop). Wait, correct phases: 0-10s (a₁ = Δv/Δt = 10/10 = 1 m/s² upward), 10-30s (a₂=0), 30-40s (a₃=Δv/Δt=(0-10)/(40-30)=-1 m/s² upward), 40-50s (a₄=Δv/Δt=(0-0)? No, 40-50s: v goes from 0 to -10? Wait no, y-axis is velocity. Let's recalculate:
Phase 1 (0-10s): v from 0→10 m/s, a₁ = (10-0)/10 = 1 m/s² (upward).
Phase 2 (10-30s): v=10 m/s (constant), a₂=0.
Phase 3 (30-40s): v from 10→0, a₃=(0-10)/10=-1 m/s² (upward, deceleration).
Phase 4 (40-50s): v from 0→-10? No, wait the graph: left side 0-10s up to 10, flat 10-30s, down 30-40s to 0, then down 40-50s to -10? But for reaction force, use Newton's 2nd law: R - mg = ma → R = m(g+a) (a upward positive).

Step2: Calculate R for each phase

Phase1: a=1 m/s², R=70(9.8+1)=7010.8=756 N.
Phase2: a=0, R=70*9.8=686 N.
Phase3: a=-1 m/s², R=70(9.8-1)=708.8=616 N.
Phase4: If v goes from 0→-10 in 10s, a=(-10-0)/10=-1 m/s² (downward, so a=-1 upward), same as phase3? Wait no, maybe phase4 is deceleration downward, but the question says "chaque phase du mouvement" so 4 phases: 0-10,10-30,30-40,40-50.

Step3: Calculate total distance (area under v-t graph)

Area = area of trapezoid or sum of shapes:
0-10s: triangle area = 0.51010=50 m.
10-30s: rectangle area=20*10=200 m.
30-40s: triangle area=0.51010=50 m.
40-50s: triangle area=0.51010=50 m (absolute value for distance).
Total distance=50+200+50+50=350 m.

Answer:

3-1: Phase 0-10s: 756 N; Phase10-30s:686 N; Phase30-40s:616 N; Phase40-50s:616 N
3-2: 350 m