Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

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.

Last Update: 2009-06-21