Describing Motion: Acceleration

Acceleration

An object moving at a constant velocity neither speeds up nor slows down. An example is a car on cruise control on a long flat stretch of road where the engine tries to apply the same force to the car. However, if a different force is applied to the object, its speed may change as it accelerates or decelerates. Just like velocity, acceleration is a vector and so direction and its corresponding signs are important.

             Mathematically, seemingly different motions are equivalent. Consider the following pairs: + acceleration and – deceleration, + deceleration and – acceleration. Think of it this way: a + acceleration is due to a force pushing against the back of the car, moving it forward to the right. Deceleration is due to a force pushing against the front of the car, slowing it. A car undergoing

– deceleration is going to the left and, as before, the – deceleration force that slows it is directed against its hood. However, realize that this direction, against the hood, is the same as that of the +force causing acceleration.

Q. Satisfy yourself that the second pair are also equivalent.

Q. Of course, the motion sensations felt by the occupants in a car during these equivalent motions

     are not equivalent. What sensations are felt during the four motions?

Average Acceleration

To find the average acceleration, one needs just the velocity change occurring during a specified time span.

a_AVE = Δ v

           Δ t

Graphical Analysis of Acceleration

As with the study of time and distance and time and velocity, there are some standard situations and graphs describing time and acceleration.

A. Time -Velocity Graphs

No Motion

The graph is a straight

horizontal line on the

x axis where the velocity is zero.

Uniform Acceleration

The graph is a straight rising

or falling line. The slope of

the line is the acceleration value itself.

Uniformly Changing Acceleration

The graph is a smooth rising or

falling arc. Tangents reveal

the instantaneous velocities

of the object at selected times.

Variably Changing Acceleration

The graph shows more variation in

pitch than that for Uniformly

Changing Acceleration. Again,

tangents reveal the instantaneous

velocities of the object at selected times.

B. Time -Acceleration Graphs

These are a more direct way of looking at acceleration than with time-velocity graphs.

No Motion

There is no time-acceleration graph for this situation. A straight horizontal line on the x axis where the acceleration is zero would mean only that the object’s speed was not changing. But, that is not the same as saying that the object’s speed was zero.

Uniform Acceleration

The graph is a straight horizontal

line intersecting the y axis at the

value of the acceleration, + or – .

Uniformly Changing Acceleration

The graph is a straight rising or falling line.

Variably Changing Acceleration

The graph is a smooth rising or falling

arc. Tangents reveal the instantaneous

accelerations of the object at selected times.

Finding Displacement During Uniform Acceleration

There are four equations used to describe the relationships among displacement, velocity, acceleration and time.

V_f = V_Ι + at d = ½ ( V_f + V_Ι )t d = V_Ι t + ½ at² V_f ² = V_Ι ² + 2ad