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: 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.

 

Example:

The graphs in figure above show the results of a fairly realistic computer simulation of the motion of a skydiver, including the effects of air friction. The x axis has been chosen pointing down, so x is increasing as she falls. Find (a) the skydiver's acceleration at t =3.0 s, and also (b) at t =7.0 s.

 

◊ The solution is shown in figure below. I've added tangent lines at the two points in question.

 

(a) To find the slope of the tangent line, I pick two points on the line (not necessarily on the actual curve): (3.0 s,28 m/s) and (5.0 s,42 m/s). The slope of the tangent line is (42 m/s-28 m/s)/(5.0 s - 3.0 s)=7.0 m/s².

(b) Two points on this tangent line are (7.0 s,47 m/s) and (9.0 s, 52 m/s). The slope of the tangent line is (52 m/s-47 m/s)/(9.0 s - 7.0 s)=2.5 m/s².

Physically, what's happening is that at t =3.0 s, the skydiver is not yet going very fast, so air friction is not yet very strong. She therefore has an acceleration almost as great as g. At t =7.0 s, she is moving almost twice as fast (about 100 miles per hour), and air friction is extremely strong, resulting in a significant departure from the idealized case of no air friction.

 

In example above, 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 below shows some examples.

 

 

 

 

Related Topics:

Acceleration

The Motion of Falling Objects

Positive and Negative Acceleration

The Area Under the Velocity-Time Graph

Algebraic Results for Constant Acceleration

Biological Effects of Weightlessness

Centrepetal Acceleration

 

Related Flash Simulations:

One-Dimensional Constant Acceleration

Velocity and Acceleratoin

Constant velocity versus constant acceleration

Freefall

Homework Problems:

Click here to get Homework Problems on acceleration.