Lectures>Explorations>Practice

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.

1. What's the object's velocity at point B?
2. How does the object's velocity at A compare to its velocity at C?
3. How is its motion changing as it goes through that time?

Interpreting a Time Graph, Part 2

The graph to the right shows the position of an object after t seconds.

1. What's the state of the object at A?
2. What's the object doing between B and C?
3. What happens to the object at D?