2D kinematics: position vector, velocity vector and acceleration vector + examples.

00:00 Introduction, how position vectors are defined in higher dimensions, trajectories and the displacement vector in two dimensions. The position is defined as a vector pointing from the origin to the location of the object, and the displacement is still the final position minus the initial position. However, the displacement is now a vector difference, and the geometric interpretation of the displacement vector is the vector pointing from the initial position to the final position.

02:49 Defining average and instantaneous velocity vectors in two dimensions. Now that we have the displacement vector, we can define the average velocity vector as the displacement vector divided by the time it takes for the object to travel between the two points. This is a vector pointing in the same direction as the displacement. Next, we can compute the instantaneous velocity by taking the small time limit of the average velocity, and we find that the instantaneous velocity vector is given by simply taking the time derivatives of the components of the position vector.

07:20 Example of calculating the average velocity vector and calculating the instantaneous velocity vector given a position vector function in terms of t. We compute average velocity as the change in the position vector divided by the change in time on the given time interval, and we compute the instantaneous velocity by differentiating the position vector function, then evaluating at the moment of interest.

09:51 Average acceleration vector and instantaneous acceleration vector. We compute the average acceleration by taking the change in velocity vector divided by the change in time. The change in velocity vector is a vector difference, so we can geometrically visualize the difference by placing the tails of the vectors in the same place, then drawing the vector from the head of the second to the head of the first vector. The average acceleration takes this same direction, since the average acceleration vector is just the velocity vector difference scaled by a factor of delta-t. To find instantaneous acceleration, we take the small time limit of this process, and we discover that the instantaneous acceleration vector is found by simply differentiating the components of the velocity vector, or by taking the second derivative of the components of the position vector.

12:40 Calculate the average and instantaneous acceleration vectors given the position vector as a function of time. We compute the average acceleration on an interval by taking the difference of velocity vectors and dividing by the size of the time interval. Next, we compute the instantaneous acceleration vector function by differentiating the components of the velocity function with respect to time, and we evaluate at the moment of interest to calculate the acceleration vector.

15:42 Review of position, velocity and acceleration functions in two dimensions. We review the formulas for position, velocity and acceleration in higher dimensions, and promise to apply our results to projectile motion in the next video!