What is Vector Acceleration in Physics? (Centripetal, Instantaneous & Average Acceleration) -[1-4-2]

The video begins with an introduction to the concept of acceleration in two and three-dimensional space. The instructor expresses excitement about explaining a fundamental physics concept that may initially seem counterintuitive: the independence of velocity and acceleration vectors in multidimensional motion. The idea is introduced that acceleration is not only about speeding up or slowing down but also about changing direction, which is a form of velocity change. The analogy of spinning an object in a circular motion is used to illustrate that even if the speed (magnitude of velocity) is constant, the direction of motion is continually changing, resulting in acceleration.

The instructor reviews the concepts of average and instantaneous velocity and acceleration in one dimension. The formulas for calculating these quantities are presented, emphasizing the difference between average velocity (delta x over delta t) and instantaneous velocity (as delta t approaches zero). Similarly, the difference between average acceleration (delta v over delta t) and instantaneous acceleration (as delta t approaches zero) is explained. The importance of understanding these concepts before extending them to two and three dimensions is highlighted.

The video continues by extending the concepts of velocity and acceleration to two and three dimensions. The average velocity is now a vector quantity, calculated as the change in position vectors over time. The instantaneous velocity is defined as the limit of this as the time interval approaches zero. Acceleration is also discussed in vector terms, with the average acceleration vector calculated as the change in velocity vectors over time. The instantaneous acceleration is introduced as the limit of delta v (change in velocity vector) over delta t as time approaches zero.

The instructor delves into the behavior of acceleration and velocity vectors in circular motion. It is shown that while the velocity vector is always tangent to the path, the acceleration vector points towards the center of the circle (centripetal acceleration). This is a key point, as it demonstrates that acceleration can occur even when there is no change in speed, only a change in direction. The concept is reinforced with a practical example of swinging an object in a circular path by a string.

The video script explains the relationship between force and acceleration, particularly in the context of circular motion. It is stated that an object in circular motion experiences a force directed towards the center of the circle, which results in centripetal acceleration. The force is provided by the tension in the string in the example, or by gravity in the case of planetary orbits. The instructor emphasizes that acceleration implies a force acting on the object and that without a force, an object would move in a straight line.

The script outlines the process of calculating the acceleration vector in circular motion. By considering the velocity at two points in time and subtracting the initial velocity vector from the final velocity vector, the change in velocity (delta v) is found. Dividing this change by the time interval (delta t) gives the average acceleration. The direction of this acceleration vector is shown to point towards the center of the circle, indicating the direction of the force causing the circular motion.

The instructor breaks down the acceleration and velocity into their x and y components to further clarify their behavior in two-dimensional space. It is shown that even though the speed (magnitude of the velocity vector) remains constant in circular motion, the x and y components of the velocity vector change, leading to a change in direction and thus acceleration. The x and y components of acceleration are also discussed, showing how they affect the respective components of velocity, resulting in the curved path of the object.

The video concludes by emphasizing that the change in direction of the velocity vector, even when the speed is constant, constitutes a change in velocity and thus results in acceleration. The instructor uses the example of an object moving in a circle to illustrate that the velocity vector's direction changes continuously, necessitating a centripetal force and acceleration. The concept is further solidified by showing how the x and y components of both velocity and acceleration interact to maintain the curved path of motion.

The final paragraph discusses the concept that acceleration can occur in a direction different from the velocity, which is a shift from the one-dimensional perspective where they are aligned. The instructor uses the example of a car accelerating sideways with a rocket thruster to illustrate how acceleration can change the direction of motion without changing its speed. This leads to a new understanding of velocity as a vector quantity that can change in both magnitude and direction due to accelerations that are not aligned with the velocity vector.