Instantaneous Velocity in Physics - Formula, Definition & Examples - [1-2-5]

The video begins with an introduction to the concept of instantaneous velocity and speed, emphasizing its importance in physics. The instructor explains that instantaneous velocity differs from average velocity as it represents the velocity of an object at a specific instant in time, rather than over a period. The lesson aims to build upon the understanding of average velocity from previous lessons and introduce the mathematical tools of calculus, specifically derivatives and limits, to calculate instantaneous velocity. The goal is to not only define instantaneous velocity but also to visualize it on a graph and calculate it using calculus.

This paragraph delves into the mathematical derivation of instantaneous velocity. The instructor explains that instantaneous velocity is found by taking the limit of the change in position (delta x) divided by the change in time (delta t) as the time interval approaches zero. This process is visually demonstrated by shrinking the time intervals on a graph until two points coincide, effectively finding the slope of the tangent line at a single point on the position-time graph. The instructor emphasizes that this slope represents the instantaneous velocity and introduces the concept of the derivative of position with respect to time, which is the fundamental idea of the lesson.

The instructor uses the example of a rocket launch to illustrate the concept of instantaneous velocity. The discussion covers how the velocity of the rocket changes over time, from being almost zero at the launch pad to increasing rapidly as it ascends. The idea of bringing two points on the graph closer together to find the velocity at a precise instant is further explained. The paragraph concludes with a graphical representation of how the velocity changes throughout the rocket's trajectory, emphasizing the importance of understanding the derivative of position with respect to time as the key to calculating instantaneous velocity.

The instructor provides a detailed explanation of how to graphically represent and calculate instantaneous velocity. Using the position-time graph, the concept of the slope of the line connecting two points and how it relates to average velocity is discussed. The process of bringing a second point closer to the first to find a better approximation of instantaneous velocity is demonstrated. The concept of a tangent line, which touches the curve at a single point, is introduced as the method to find the instantaneous velocity. The paragraph concludes with the mathematical representation of instantaneous velocity as the limit of delta x over delta t as delta t approaches zero, and the renaming of this process in calculus as dx/dt.

This paragraph focuses on the direction of velocity and introduces the concept of instantaneous speed. The instructor explains that velocity is a vector quantity, meaning it has both magnitude and direction, while speed is the magnitude of velocity without regard to direction. The discussion includes how to determine the sign of velocity based on the slope of the tangent line at a point on the position-time graph. The paragraph also covers how to calculate instantaneous speed by taking the absolute value of the velocity. The importance of understanding the sign of velocity in determining the direction of motion is emphasized.

The instructor presents a practical example of calculating instantaneous velocity using a given position function. The example involves a particle moving along a parabolic path, and the position function is given as x = t^2. The derivative of this function with respect to time is calculated to find the velocity function, which is 2t. The velocity at various time intervals is then determined and tabulated. The paragraph concludes with a discussion on how the velocity increases over time and how this is reflected in the graph of the velocity function, showing a constant rate of acceleration.

The final paragraph summarizes the key points of the lesson, reinforcing the understanding of instantaneous velocity as the derivative of the position function with respect to time. The process of finding instantaneous velocity by taking the limit of delta x over delta t as the time interval becomes infinitesimally small is reiterated. The paragraph also reviews the concepts of average velocity, the graphical representation of velocity through the slope of the tangent line, and the distinction between velocity and speed. The instructor encourages further practice and application of these concepts in future lessons to solidify understanding and apply them to problems in physics.