3.3
La vélocité instantanée - I
La vélocité moyenne pendant un intervalle de temps ne peut pas nous dire à quelle vitesse ou dans quelle direction une particule se déplace à un momen…
La vitesse moyenne d’un objet ne peut pas nous dire à quelle vitesse ou dans quelle direction l’objet se déplaçait à un moment donné. Pour l’identifier, nous avons besoin du vecteur vitesse à un instant précis ou du vecteur vitesse instantané.
Le vecteur vitesse instantanée est la limite du vecteur vitesse moyen lorsque l’intervalle de temps tend vers zéro, ou la dérivée de x par rapport à t. Ici, la variable x est utilisée pour décrire la position d’un objet le long d’un mouvement unidimensionnel. La vitesse instantanée est une grandeur vectorielle, et le symbole vx est utilisé pour la vitesse instantanée le long de la direction x.
Le signe de la vitesse instantanée est le même que le signe de Δx, car le temps est toujours considéré comme positif. Par exemple, si x augmente et que le mouvement est dans la direction positive x, cela implique une valeur positive de vitesse instantanée. De même, si x diminue et que le mouvement est dans la direction négative x, la vitesse instantanée a une valeur négative.
View the full transcript and gain access to JoVE Core videos
Q1: Why is instantaneous velocity different from average velocity?
Average velocity cannot reveal how fast or in what direction an object moves at any specific moment. Instantaneous velocity measures the velocity at a single instant in time, providing precise information about motion at that exact point. It is the limit of average velocity as the time interval approaches zero, giving a complete picture of motion dynamics.
Q2: How is instantaneous velocity mathematically defined?
Instantaneous velocity is defined as the limit of average velocity as the time interval approaches zero, or equivalently, the derivative of position x with respect to time t. This mathematical definition allows us to calculate the exact velocity at any specific instant. The symbol vx represents instantaneous velocity along the x-direction in one-dimensional motion.
Q3: What does the sign of instantaneous velocity indicate?
The sign of instantaneous velocity indicates the direction of motion. A positive instantaneous velocity means position x is increasing and motion is in the positive x-direction. A negative instantaneous velocity means position x is decreasing and motion is in the negative x-direction. Time is always positive, so the sign depends entirely on the change in position.
Q4: Is instantaneous velocity a scalar or vector quantity?
Instantaneous velocity is a vector quantity, meaning it has both magnitude and direction. It has dimensions of length per unit time and can be positive or negative depending on the direction of motion. This distinguishes it from instantaneous speed, which is the magnitude of instantaneous velocity and is always positive.
Q5: How do instantaneous velocity and instantaneous speed differ?
Instantaneous velocity is a vector with direction and sign, while instantaneous speed is the magnitude of instantaneous velocity and is always positive. For example, particles with velocities of −10 m/s and +10 m/s have the same instantaneous speed but travel in opposite directions. Speed tells us how fast an object moves; velocity tells us how fast and in which direction.
Q6: What role does the time interval play in calculating instantaneous velocity?
Instantaneous velocity is found by taking the limit of average velocity as the time interval approaches zero. As the time interval becomes infinitesimally small, the average velocity converges to the instantaneous velocity at that specific moment. This limiting process is the foundation of the derivative, which mathematically defines instantaneous velocity.
Q7: Can you determine instantaneous velocity from a position-time graph?
Yes, instantaneous velocity can be determined from a position-time graph using the velocity and position by graphical method. The instantaneous velocity at any point equals the slope of the tangent line to the curve at that point. This graphical approach provides a visual way to understand how velocity changes throughout an object's motion.