3.8
Équations cinématiques - II
La deuxième équation cinématique exprime la position finale d'un objet en fonction de sa position initiale, de la distance parcourue avec une vélocité…
La première équation cinématique exprime le vecteur vitesse en fonction du vecteur vitesse initial et de la variation du vecteur vitesse.
Pour calculer la deuxième équation cinématique, deux équations pour la vitesse moyenne x sont utilisées pendant un intervalle de t égal à zéro à un temps ultérieur t. La première équation de la vitesse moyenne x est la variation du déplacement au cours de la variation du temps, comme nous l’avons vu dans les leçons précédentes. Ici, la position initiale est notée x0 à l’instant t égal à zéro et x à l’instant t ultérieur.
En supposant que l’accélération est constante , le vecteur vitesse x moyen peut également être représenté comme la moyenne des vitesses aux temps initial et final. En substituant ici la première équation cinématique, on obtient une équation pour la position x en fonction du temps.
Cette équation montre que la position d’un objet à n’importe quel instant t est la somme de sa position initiale, de la distance parcourue sous une vitesse initiale constante et de la distance parcourue lors du changement de vitesse.
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Q1: What does the second kinematic equation describe?
The second kinematic equation expresses an object's final position as the sum of its initial position, the distance moved under constant initial velocity, and the distance traveled due to acceleration. This equation applies only when acceleration remains constant throughout the motion, allowing you to calculate position at any time t.
Q2: How is the second kinematic equation derived?
The derivation uses two expressions for average velocity over a time interval. The first defines average velocity as change in displacement over change in time. The second represents average velocity as the mean of initial and final velocities. Substituting the first kinematic equation into this relationship yields the position equation.
Q3: What are the components of position in the second kinematic equation?
Position consists of three components: the initial position x0, the distance traveled at constant initial velocity, and the distance covered during the velocity change caused by acceleration. Together, these components determine where an object is located at any given time t during constant acceleration motion.
Q4: How do you solve for time using the second kinematic equation?
Rearranging the second kinematic equation produces a quadratic equation in time. Using the quadratic formula yields two mathematical solutions. However, only the positive solution is physically meaningful, since negative time would indicate an event occurring before motion began, making it unreasonable to accept.
Q5: Can you apply the second kinematic equation to a freeway merge scenario?
Yes. For a car merging onto a 200 m ramp with initial velocity 10 m/s and acceleration 2 m/s², substitute these known values into the second kinematic equation. Solving the resulting quadratic equation gives t = 10 s as the valid solution, meaning the car takes 10 seconds to travel the ramp.
Q6: Why is constant acceleration required for the second kinematic equation?
The second kinematic equation is derived assuming constant acceleration, which allows average velocity to be expressed as the mean of initial and final velocities. If acceleration varies, this relationship breaks down, and the equation no longer accurately predicts position, making constant acceleration a fundamental requirement.
Q7: How does the second kinematic equation relate to other kinematic concepts?
The second kinematic equation builds on the first kinematic equation, which relates velocity to acceleration. Together with concepts like average velocity and instantaneous velocity, these equations form a complete framework for analyzing motion with constant acceleration and solving kinematic equations problem solving scenarios.