Varying Acceleration

So far we have only been discussing examples of motion for which the v-t graph is linear. If we wish to generalize our definition to v-t graphs that are more complex curves, the best way to proceed is similar to how we defined velocity for curved x-t graphs:

definition of acceleration

The acceleration of an object at any instant is the slope of the tangent line passing through its v-versus-t graph at the relevant point.

A skydiver.

In the skydiver example, the x-t graph was not even used in the solution of the problem, since the definition of acceleration refers to the slope of the v-t graph. It is possible, however, to interpret an x-t graph to find out something about the acceleration. An object with zero acceleration, i.e., constant velocity, has an x-t graph that is a straight line. A straight line has no curvature. A change in velocity requires a change in the slope of the x-t graph, which means that it is a curve rather than a line. Thus acceleration relates to the curvature of the x-t graph. Figure m shows some examples.

In the example 6, the x-t graph was more strongly curved at the beginning, and became nearly straight at the end. If the x-t graph is nearly straight, then its slope, the velocity, is nearly constant, and the acceleration is therefore small. We can thus interpret the acceleration as representing the curvature of the x-t graph, as shown in figure m. If the "cup" of the curve points up, the acceleration is positive, and if it points down, the acceleration is negative.

m / Acceleration relates to the curvature of the x-t graph.

Since the relationship between a and v is analogous to the relationship between v and x, we can also make graphs of acceleration as a function of time, as shown in figure n.

n / Examples of graphs of x, v, and a versus t. 1. A object in free fall, with no friction. 2. A continuation of example 6, the skydiver.

→ Solved problem: Drawing a v-t graph. page 125, problem 14

→ Solved problem: Drawing v-t and a-t graphs. page 126, problem 20

o / How position, velocity, and acceleration are related.

Figure o summarizes the relationships among the three types of graphs.

Discussion Questions

A	Describe in words how the changes in the a-t graph in figure n/2 relate to the behavior of the v-t graph.
B	Explain how each set of graphs contains inconsistencies, and fix them.
C	In each case, pick a coordinate system and draw x - t, v - t, and a-t graphs. Picking a coordinate system means picking where you want x = 0 to be, and also picking a direction for the positive x axis. (1) An ocean liner is crusing in a straight line at constant speed. (2) You drop a ball. Draw two different sets of graphs (a total of 6), with one set's positive x axis pointing in the opposite direction compared to the other's. (3) You're driving down the street looking for a house you've never been to before. You realize you've passed the address, so you slow down, put the car in reverse, back up, and stop in front of the house.