3-E Free Fall
Instructions
The applet shows the motion of a point along a vertical (one-dimensional) line.
- Notice that the (instantaneous) position, velocity, and acceleration of the point are given.
- Use the slider tool or input box to move the point P to a specific time value.
- Use the check boxes to show/hide the tangent line, graph of , and graph of .
3-E Higher Order Derivatives and Applications
The derivative of a function is the instantaneous rate of change. For example, the (first) derivative of position is velocity and describes how quickly (or slowly) the position is changing at any given time. But, velocity itself is a function and has its own rate of change, which we call acceleration. In other words, acceleration describes how the velocity changes, which in turn describes how the position changes. So, acceleration also tells us something about position.
More generally, a function has a derivative, and the derivative has a derivative (and that derivative has a derivative), and so on. The order of a derivative is essentially how many times we differentiated the "original" function. So, the second derivative is the derivative of the (first) derivative. The third derivative is the derivative of the second derivative, and so on.
Free Fall: A moving object is considered to be in free fall when the only force acting on it is gravity. For objects near the surface of the earth there is a constant downward acceleration of magnitude 9.8 m/s per second. Why are these problems typically modeled by parabolas? What type of functions would have a constant acceleration (i.e., constant second derivative)?