2.2 - Instantaneous Velocity

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00:00 Intro
Average velocity between two points:

v_av−x=ΔxΔt= (x2−x1)/(t2−t1)

Finding the velocity at a single instant is not actually a simple problem to solve.

04:30 Getting to Instantaneous Velocity
A rocket ship that starts out stationary and after a few seconds it rapidly picks up speed. The average velocity of the first 10 seconds would be a poor estimate of the actual instantaneous velocity at t=10s. This is because the rocket continues to get faster and faster over time. By averaging in very slow velocities early on, we will severely underestimate the actual instantaneous velocity at t=10s.

To resolve this issue we calculate an average velocity using a smaller time interval. Instead of calculating the average velocity over the first 10s, we use the last second, or even the last tenth of a second.

12:42 Limits
Using smaller and smaller Δt intervals ignores sections of the rocket’s motion where the velocity is very different from t=10s. As the Δt interval goes to zero, our average velocity calculation approaches the value of the instantaneous velocity.

Instantaneous Velocity=v_x = lim Δt→0 Δx/Δt = dx/dt

14:20 Derivatives
Where dx/dt is the derivative of x with respect to t. It represents the instantaneous rate of change of x with respect to t at any specific time.

16:20 Power Rule of Differentiation
The most common way to compute derivatives in this course is the power rule. Assume we have an expression for the position as a function of time:

x=at^n
where a is some proportionality constant. In this case the derivative of x with respect to time (or the rate of change of x over time) is

dx/dt = n×at^(n−1)

18:10 Alternate Derivative Notation
dx/dt can also be written as x'(t) or ẋ

20:02 Sign of Velocity
The sign of v_x will be equal to the sign of Δx/Δt. The time interval Δt will always be positive because time only moves forwards. Thus the sign of the velocity must equal the sign of the displacement Δx.

23:48 Velocity vs Speed
Speed and Velocity mean different things in physics. Speed is a scalar, while velocity is a vector.

speed=distance/time
As a scalar it cannot be negative. It is only possible to traverse positive (or zero) distance.

velocity=displacement/time
Where displacement is a vector with direction that can be positive or negative. Thus velocity is also a vector and can be positive or negative.

Average speed and average velocity are also not the same quantity. For example for a person walking in a circle, their total displacement is zero if they return to their original position, thus average velocity = 0. However average speed will be some positive number because distance/time ≠ 0.

For instantaneous speed, we will use the symbol v.

For instantaneous velocity, we will often use symbols like vx,vy,vz or we can use instantaneous speed plus a direction.

29:15 Position - Time Graph
If we draw a straight line connecting two points on a position-time graph, then the slope of the line is equal to the average velocity between those two points.

35:15 Tangent Lines
The slope of the tangent line on a position-time graph represents the instantaneous velocity of the object at that point.

44:16 Interpreting x-t and Motion Diagrams
By looking at the slope of the curve on a position-time graph at any point, we can see both the direction and magnitude of the object's velocity. We can translate this into a snapshot of the system using a motion diagram.