Math Problem Statement
Un mobile M (une voiture) se déplace suivant un axe X’OX. Son digramme de vitesse est donné sur la figure ci-dessous. On suppose qu’à l’instant t=0 (s), x(0)= -1 (m). A partir du diagramme :
- Calculer les positions du mobile aux instants 1(s) et 4 (s).
- Retrouver la distance parcourue pendant les quatre premières secondes. Quelle est sa vitesse scalaire moyenne.
- Calculer la vitesse moyenne entre les instants 1(s) et 4 (s).
- Déterminer son accélération instantanée à l’ instant t= 1(s)
- Déterminer son accélération moyenne sur l’intervalle [1s, 5s].
- Pour décrire le mouvement du mobile, préciser : a- La vitesse à l’instant initial t=0 (s). b- Le sens de son mouvement à l’instant initial. c- L’instant et la position auxquelles il rebrousse le chemin.
- Donner les différentes phases du mouvement et leurs natures
- Retrouver l’équation horaire du mouvement et tracer le digramme des espaces.
Solution
To tackle this problem, we need to carefully analyze the velocity-time graph (speed diagram) and use it to find the required kinematic information. Here's a structured approach to solve each part:
Given Data:
- : m
- We’ll interpret specific values from the velocity-time graph to solve the problem.
Solutions:
1. Calculating the Position of the Mobile at s and s
To find the position at s and s, we use the relationship between velocity and displacement: We need to interpret the area under the velocity curve between s and s, and between s and s. Positive areas represent displacement in the positive -direction, while negative areas indicate displacement in the opposite direction.
2. Total Distance Traveled and Average Scalar Speed in the First 4 Seconds
The distance is the sum of absolute values of the areas under the curve from s to s. To find the average speed, use:
3. Average Velocity Between s and s
The average velocity over the interval from ( t = 1 \
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Math Problem Analysis
Mathematical Concepts
Kinematics
Velocity and Displacement
Acceleration
Integral Calculus
Formulas
x(t) = x(0) + ∫(0 to t) v(t) dt
Average Speed = Total Distance / Total Time
Average Velocity = Δx / Δt
Theorems
Fundamental Theorem of Calculus (used to find displacement from velocity)
Definition of Average Speed and Average Velocity
Suitable Grade Level
Grades 11-12
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