We have discussed displacement and velocity. Recall that we defined velocity as a change of displacement per unit time. This could be as a result of a change in an object's speed, direction, or both. Now, we will define a new term: acceleration. Average acceleration (symbol 'a') is a change in velocity per unit time, or
Acceleration is a vector quantity - it has both magnitude and direction. We can accelerate an object by changing its speed over a time interval, such as speeding up or slowing down in your car. We are familiar with the right hand gas pedal on a car - we call it the "accelerator." But the brake pedal can slow us down so it can also be called an "accelerator". We can also change the velocity of an object by changing the direction the object is moving. So in a car, the steering wheel is also a type of "accelerator".
The units of acceleration are units of velocity per unit time. Usually, we will use our standard SI units, so we would have (meters per second) per second, or m/s/s. We commonly write this as m/s^{2}. For example, if a car accelerates from rest to 20 m/s in 5 seconds, then its acceleration is given by
The direction of acceleration is generally given in terms of the sign of the displacement and the velocity. Again, we will use positive and negative signs to show the direction of our acceleration. What we choose as positive (left, right, up, down, etc) is largely a matter of personal preference, but we will see in some later problem solving examples that sometimes it makes sense to choose our sign conventions in a specific way. We can make some interesting observations which may conceptually help us understand signs. An object may be moving in a forward direction (+ velocity) and have negative acceleration. An example of this is a car with its brakes on. An object may be at rest (zero velocity) and have positive acceleration (such as a car just starting from rest; it is the acceleration on the car that causes a change in velocity.) There are other examples that we will discuss in future lessons.
Motion with Uniform Acceleration
The next thing to consider is a specialized case of acceleration which we will study for quite some time: motion in a straight line (direction is not changing) with a uniform change in speed. We sometimes call this uniformly accelerated motion. Here are the assumptions for our next steps:
- An object is moving in a straight line
- The rate of change of velocity is uniform (the same).
Let's let our initial time, t_{o} = 0, so that in any case, Dt=t_{f}=t. Our initial position and velocity will be referred to as x_{o} and v_{o}. Finally, let x = final position (x_{f}) and v = final velocity (v_{f}). Now we are ready to develop the basic equations of linear motion.
From the definition of acceleration:
If we want to solve for velocity at any time, we can rearrange the terms so that
This is our first equation. Now recall from our lesson on velocity, that average velocity (using the terms we defined above and valid only with uniform acceleration) may be expressed as both
and
The second equation may be rewritten as x = x_{o} + v_{ave}t. This may further be rewritten as
and substituting in
from above, we obtain
or
which is an incredibly useful equation for determining position as a function of time. Finally, if we take the equation v=v_{o} +at and solve for t, we get
Substituting this equation into
and solving for v^{2}, we get yet another equation for motion:
The equations derived (and shown again to the right in the form that they appear in our classroom) are four general motion equations that will allow us to solve various problems for displacement, velocity, acceleration, and time based on information we are given. We will apply these in later lessons. All of these are only valid, however, for the case where acceleration is a uniform value.
Just as we graphed displacement vs. time and velocity vs. time, let's also consider the graph of acceleration vs. time for the idea of uniform acceleration. The data for this graph generally starts with a velocity vs. time graph, so let's consider the following example:
- In our graph above (region 1), we start with a velocity of 0 m/s (at rest) and accelerate to 10 m/s in 10 seconds.
- In region 2, we remain at 10 meters per second for 10 more seconds.
- In region 3, we accelerate to 20 meters per second in 4 seconds.
- In region 4, we negatively accelerate to -10 m/s in 10 seconds.
- In region 5 we stay at -10 m/s.
In a similar manner from our lesson on velocity graphs, we'll look at the slope of the line in each region. Here, our slope (rise over run) is the change in velocity divided by the change in time, or Dv/ D t. This is our definition of acceleration. So, the slope of a velocity vs. time graph is the acceleration of the object at that interval. What are our slopes?
Interval Number |
Slope = acceleration |
1 |
D = 10 m/s in 10 sec = 1 m/s^{2} |
2 |
D = 0 m/s in |
3 |
D = 10 m/s in |
4 |
D = -30 m/s |
5 |
D = 0 m/s in |
So what does the graph look like?
Note that each of the lines above is a horizontal (or zero slope) line. This means that the acceleration is not changing during the intervals. It appears to be making step changes from one interval to the next, but in reality, there would be some line that connects each interval since it is really hard to have an instantaneous change in acceleration from one number to the next.
A check to see if you have done the reading: For extra credit in class, on a sheet of paper, draw the above graphs and show me the times that the velocity of this object is zero, and compute for me the total displacement of the object.
Often it is helpful to see what someone else has to say on the topic. Try http://www.physicsclassroom.com/Class/1DKin/U1L1e.html for more information on acceleration.
For Practice Problems, Try: From the University of Oregon, any of these:
Giancoli Multiple Choice Practice Questions (Questions 1-12)