Overview: Considering that you’re a calculus student, you should be sickeningly familiar with the notion of "rate of change." Nevertheless, this important concept manifests in a number of contexts in math, and is not to be taken lightly. Looking at various graphs of time versus position, one can intuitively see that slope plays an important role in the direction in which the object is moving. Before we continue, take a look at the three graphs below - consider how slope plays a role in each one.
Concepts: As you’ve just seen, if the position function has a positive slope at a time ‘t,' then the object is moving in the right direction of the origin, and if the slope is negative, then the object is moving in the left direction. If the velocity is zero, then the object is at rest. Bear in mind that an object could be located to the right of the origin, yet still be moving in the left direction, and vice versa. With these concepts in mind, I encourage you to reconsider the graphs above! Show/Hide Graph Answers.
- Graph 1:The movement to the right corresponds to the positive slope, and the "break" corresponds to the slope of zero.
- Graph 2: The consistent movement to the left corresponds to the consistent negative slope.
- Graph 3: The movement to the right corresponds to the positive slope. Then, the "break" corresponds to the slope of zero, and finally the movement to the left corresponds to the negative slope.
Though we’ve discussed the intuitive nature of velocity, it’s still necessary to go over the formalities: velocity is the rate at which position is changing. Thus, by taking the derivative of the position function, we can find the velocity of the object at any time 't.' So, let’s write this as d’(t) = v(t). Moreover, since we also know that critical values of a function are the locations where slope is possibly changing its sign value, we can deduce that if v(t) = 0, this means that the object is either about to change directions (from moving left to moving right, or vice versa), or it is remaining still for a while.
Practice - analyze the three functions of position or velocity for the separate objects, and then consider which statement accurately describes the object's motion. Once you've figured it out, mouse over the statement to see if you're correct!
Object 1: d(t) = (1⁄8)t2 - 2t + 2 Object 2: v(t) = t2 – 16t + 64 Object 3: d(t) = 6
Speed: Before moving on, it’s necessary to formalize the notion of "speed." While one may think that velocity and speed are synonymous, they’re actually distinct terms in physics. Speed is the absolute value of velocity, meaning that if s(t) is the speed function, then s(t) = |v(t)|. As an example, if v(t) = -2t, then s(t) = |-2t| = 2t. For an intuitive explanation of this, think about a scenario where a car is going down a straight, two-lane highway. If the car is going in the negative direction, but at a high velocity, they would still get pulled over for speeding. Hence, speed is important because it gives us the magnitude of velocity.