The final mathematical quantity discussed in Lesson 1
is acceleration. An often misunderstood quantity, acceleration
has a meaning much different from the meaning sports announcers and other individuals
associate with it. The
definition of acceleration is:
Acceleration
is a vector quantity which is
defined as "the rate at which an object changes its
velocity." An object is
accelerating if it is changing its velocity.
Sports
announcers will occasionally say that a person is
accelerating if he/she is moving fast. Yet acceleration
has nothing to do with going fast. A person can be moving
very fast, and still not be accelerating. Acceleration
has to do with changing how fast an object is moving. If
an object is not changing its velocity, then the object
is not accelerating. The data at the right is
representative of an accelerating object – the velocity
is changing with respect to time. In fact, the velocity
is changing by a constant amount - 10 m/s - in each
second of time. Whenever an object's velocity is changing,
that object is said to be accelerating; that object has an
acceleration.
Constant Acceleration
Sometimes an accelerating object will change its
velocity by the same amount each second. As mentioned
before, the data above shows an object
changing its velocity by 10 m/s in each consecutive
second. This is known as a
constant acceleration
since the velocity is changing by the same amount each
second. An object with a constant acceleration should not
be confused with an object with a constant velocity.
Don't be fooled! If an object is changing its velocity
– whether by a constant amount or a varying amount –
it is an accelerating object. An object with a
constant velocity is not accelerating. The data tables
below depict motions of objects with a constant
acceleration and with a changing acceleration. Note that each
object has a changing velocity.
Since
accelerating objects are constantly changing their
velocity, you can say that the distance traveled divided
by the time taken to travel that distance is
not a constant value. A falling object for instance
usually accelerates as it falls. If you were to observe
the motion of a free-falling
object (free fall
motion will be discussed in detail later), you would
notice that the object averages a velocity of 5 m/s in
the first second, 15 m/s in the second second, 25 m/s in
the third second, 35 m/s in the fourth second, etc. Our
free-falling object would be accelerating at a constant rate.
Given these average velocity values during each
consecutive 1-second time interval,
the object falls:
– 5 meters in the first second,
– 15 meters in the second second (for a total distance of 20 meters),
– 25 meters in the third second (for a total distance of 45 meters),
– 35 meters in the fourth second (for a total
distance of 80 meters).
These numbers are summarized in the table below.
Time
Interval
Average
Velocity During Time
Interval
Distance
Traveled During Time
Interval
Total
Distance Traveled from 0 s to End of Time
Interval
0 - 1 s
5 m/s
5 m
5 m
1 - 2 s
15 m/s
15 m
20 m
2 - 3
25 m/s
25 m
45 m
3 - 4 s
35 m/s
35 m
80 m
This discussion illustrates that a
free-falling object which is
accelerating at a constant rate will cover different
distances in each consecutive second. Further analysis of
the first and last columns of the table above reveal that
there is a square relationship between the total distance
traveled and the time of travel for an object starting
from rest and moving with a constant acceleration.
For objects with a constant acceleration, the
distance of travel is directly proportional to the square
of the time of travel.
As such, if an object travels for
twice the time, it will cover four times (22) the
distance; the total distance traveled after two seconds
is four times the total distance traveled after one
second.
If an object travels for three times the time,
then it will cover nine times (32) the distance; the
distance traveled after three seconds is nine times the
distance traveled after one second.
Finally, if an object
travels for four times the time, then it will cover sixteen
times (42) the distance; the distance traveled after
four seconds is sixteen times the distance traveled after one
second.
Calculating Acceleration
The acceleration of any object is
calculated using the equation:
This equation can be used to calculate the
acceleration of the object whose motion is depicted by
the velocity-time data table
above. The velocity-time data in the table shows that the
object has an acceleration of 10 m/s/s. The calculation
is shown below:
Acceleration
values are expressed in units of velocity per time. Typical
acceleration units include the following:
m/s/s
mi/hr/s
km/hr/s
Initially, these units are a little
awkward to the newcomer to physics. Yet, they are very
reasonable units when you consider the
definition of and equation for acceleration. The reason for
the units becomes obvious upon examination of the
acceleration equation.
Since acceleration is a velocity change over a time interval,
the units for acceleration are velocity units divided by
time units – thus (m/s)/s or (mi/hr)/s.
Direction of the Acceleration Vector
Acceleration is a vector
quantity so it will always have a direction associated
with it. The direction of the acceleration vector depends
on two factors:
whether the object is speeding up or slowing
down
whether the object is moving in the positive (+) or negative (–)
direction
The general
RULE OF THUMB is:
If an object is slowing down, then its
acceleration is in the opposite direction of its
motion.
This RULE OF THUMB can be
applied to determine whether the sign of the acceleration
of an object is positive or negative, right or left, up
or down, etc. Consider the two data tables below.
In Example A, the object is moving in the
positive direction (i.e., has a positive velocity) and is
speeding up. When an object is speeding up, the
acceleration is in the same direction as the velocity.
Thus, this object has a positive
acceleration.
In Example B, the object is
moving in the negative direction (i.e., has a negative
velocity) and is slowing down. When an
object is slowing down, the acceleration is in the
opposite direction as the velocity. Thus, this object
also has a positive
acceleration.
This same RULE OF
THUMB can be applied to the motion of the objects
represented in the two data tables below.
In Example C, the object is moving in the
positive direction (i.e., has a positive velocity) and is
slowing down. When an object is slowing down, the
acceleration is in the opposite direction as the
velocity. Thus, this object has a
negative
acceleration.
In Example D, the object is
moving in the negative direction (i.e., has a negative
velocity) and is speeding up. When an object is speeding
up, the acceleration is in the same direction as the
velocity. Thus, this object also has a
negative
acceleration.
Check
Your Understanding
To test your understanding of the concept of
acceleration, consider the following problems and their
corresponding solutions. Use the equation
to determine the acceleration for the
two motions below.
