Acceleration  

Acceleration is one of those words for which we have a common meaning and a meaning that is specific to the field of physics. Ask the average person on the street and they will define acceleration as "speeding up." However, this is just a portion of the full meaning of acceleration as used in physics. In physics, acceleration is defined as a change in velocity. Considering this definition closely, we remember that velocity is defined as speed in a given direction. Therefore changing the velocity of an object means changing its speed (rate of motion), its direction of travel or both variables. An object experiencing a change of speed or direction is said to be accelerating.

Mathematically, we can express acceleration with the following formula:

a = (vf - vo)/t

where:

a - acceleration
vf - final velocity
vo - original velocity
t - time


The value for acceleration may be a negative or positive value depending on magnitude of vf and vo, but it is still properly termed acceleration (although we do often call negative acceleration "deceleration").

The unit for acceleration takes the form of

velocity/time2

where the time unit for velocity matches the time unit in the denominator of the acceleration formula. For example:

m/sec2
km/hr2
cm/min2

The derivation of this unit makes sense if it is broken down into simple steps:

Let's report our velocity measurements in m/sec and our time unit in seconds. When we solve the numerical portion of our equation we are left with:

m/sec
___________________
sec

Remembering back to basic math, when we divide a number by another, it is the same as multiplying by the inverse of the second number. For instance:

3/4 = 3 x 1/4

By the same principle:

m/sec/sec = m/sec x 1/sec

The resulting unit would therefore be m/sec2.

Sample Problem
A car starts from a stoplight and is traveling with a velocity of 10 m/sec east in 20 seconds. What is the acceleration of the car?

First we identify the information that we are given in the problem:
vf - 10 m/sec
vo - 0 m/sec
time - 20 seconds
Then we insert the given information into the acceleration formula:
a = (vf - vo)/t
a = (10 m/sec - 0 m/sec)/20 sec
Solving the problem gives an acceleration value of 0.5 m/sec2.
Sample Problem
A ball is rolled at a velocity of 12 m/sec. after 36 seconds, it comes to a stop. What is the acceleration of the ball?
First we identify the information that we are given in the problem:
vf - 0 m/sec
vo - 12 m/sec
time - 36 seconds
Then we insert the given information into the acceleration formula:
a = (vf - vo)/t
a = (0 m/sec - 12 m/sec)/36 sec
Solving the problem gives an acceleration value of -0.33 m/sec2.

Sometimes we already know the acceleration of an object and want to calculate some other aspect of the object's motion such as its original or final velocity or the time it took for the change in velocity to occur. The same formula allows us to calculate any missing variable.
Sample Problem
A falling object is accelerated by gravity at 9.8 m/sec2. If it takes 15 seconds for the object to hit the ground after it has been released, with what velocity will the object impact the ground?

First we identify the information that we are given in the problem:
a - 9.8 m/sec2
vo - 0 m/sec
time - 15 seconds
Then we insert the given information into the acceleration formula:
a = (vf - vo)/t
9.8 m/sec2 = (vf - 0 m/sec)/15 sec
Note that the formula may be rewritten:
9.8 m/sec2/1 = vf/15 sec
With equivalent fractions, the first step towards the solution is to cross multiply: 9.8 m/sec2 x 15 sec = 1 x vf
vf = 147 m/sec

Graphically, acceleration appears different than speed. Constant speed assumes a straight line and average speed may take any shape on a line graph, depending on the actual motion of the object. Acceleration appears as an upward or downward curved line:


In the above graph, for each equal unit of time, the object is covering a greater amount of distance. This is the nature of acceleration.