Instantaneous Velocity
We talked briefly about average versus instantaneous velocity in the lectures but it's an important application that we can learn a lot from so it's worth looking at in more detail. Remember that, if the position of an object at time t is given by the function $s(t)$ then it's instantaneous velocity at $t=a$ is
$$v(a) = s'(a) = \lim_\limits{x \to a}\frac{f(x+a)-f(a)}{h}$$To interpret this, remember what velocity is: a measure of how fast the distance traveled by an object is changing. So, a positive slope means the object is moving in a positive direction and a negative slope means the object is moving in a negative direction. Finally, the greater the derivative, the steeper the slope and the faster the object is moving.
Interpreting a Time Graph, Part 1
The graph to the right shows the position of an object after t seconds. Don't confuse this with a graph that shows the actual position of the object. Think of the object as moving along a number line. At $t=0$ the point is at the origin. At $t=1$ the point is 2.5 units to the right, at $t=2$ it's at 4 units to the right then at $t=3$ it's back to $x=2.5$, etc.
- What's the object's velocity at point B?
- How does the object's velocity at A compare to its velocity at C?
- How is its motion changing as it goes through that time?
The tangent line at point B is horizontal which means its slope is 0. Since the slope represents the object's velocity, its velocity at that point is 0 which means the object has stopped there.
The magnitude of the slope is the same on both sides but the sign is different, positive on the left and negative on the right. THat means that the object's velocity is positive on the left side of B and negative on the right side.
Since the function's velocity (slope) is positive on the left side of B and negative on the right side, so the object must have stopped at point B and turned around.
Interpreting a Time Graph, Part 2
The graph to the right shows the position of an object after t seconds.
- What's the state of the object at A?
- What's the object doing between B and C?
- What happens to the object at D?
The slope of the tangent line at this point is positive so the object is already moving in the positive x direction at $t=0$.
The tangent line has a slope of 0 through this entire section. Since it's velocity is 0, the object isn't moving between these points.
The tangent line is horizontal here so the velocity is 0 and the object has stopped moving again. Since the slope is negative on one side and positive on the other, the object has turned around and is moving back toward $x=0$.
