Position, Velocity, and Acceleration in One Dimension
We have already discussed examples of position functions in the previous section. We now turn our attention to velocity and acceleration functions in order to understand the role that these quantities play in describing the motion of objects. We will find that position, velocity, and acceleration are all tightly interconnected notions.
Velocity in One Dimension
In one dimension, velocity is almost exactly the same as what we normally call speed. The speed of an object (relative to some fixed reference frame) is a measure of "how fast" the object is going--and coincides precisely with the idea of speed that we normally use in reference to a moving vehicle. Velocity in one-dimension takes into account one additional piece of information that speed, however, does not: the direction of the moving object. Once a coordinate axis has been chosen for a particular problem, the velocity v of an object moving at a speed s will either be v = s , if the object is moving in the positive direction, or v = - s , if the object is moving in the opposite (negative) direction.
More explicitly, the velocity of an object is its change in position per unit time, and is hence usually given in units such as m/s (meters per second) or km/hr (kilometers per hour). The velocity function, v(t) , of an object will give the object's velocity at each instant in time--just as the speedometer of a car allows the driver to see how fast he is going. The value of the function v at a particular time t 0 is also known as the instantaneous velocity of the object at time t = t 0 , although the word "instantaneous" here is a bit redundant and is usually used only to emphasize the distinction between the velocity of an object at a particular instant and its "average velocity" over a longer time interval. (Those familiar with elementary calculus will recognize the velocity function as the time derivative of the position function.)
Average Velocity and Instantaneous Velocity
Now that we have a better grasp of what velocity is, we can more precisely define its relationship to position.
We begin by writing down the formula for average velocity. The average velocity of an object with position function x(t) over the time interval (t 0, t 1) is given by:
In other words, the average velocity is the total displacement divided by the total time. Notice that if a car leaves its garage in the morning, drives all around town throughout the day, and ends up right back in the same garage at night, its displacement is 0, meaning its average velocity for the whole day is also 0.
As the time intervals get smaller and smaller in the equation for average velocity, we approach the instantaneous velocity of an object. The formula we arrive at for the velocity of an object with position function x(t) at a particular instant of time t is thus:
This is, in fact, the formula for the velocity function in terms of the position function! (In the language of calculus, this is also known as the formula for the derivative of x with respect to t .) Unfortunately, it is not feasible, in general, to compute this limit for every single value of t. However, the position functions we will be dealing with in this SparkNote (and those you will likely have to deal with in class) have exceptionally simple forms, and hence it is possible for us to write down their corresponding velocity functions in terms of a single rule valid for all time. In order to do this, we will borrow some results from elementary calculus. These results will also prove useful in our discussion of acceleration.
Some Useful Results from Elementary Calculus
Loosely speaking, the time derivative of a function f (t) is a new function f'(t) that keeps track of the rate of change of f in time. Just as in our formula for velocity, we have, in general:
Notice that this means we can write: v(t) = x'(t) . Similarly, we can also take the derivative of the derivative of a function, which yields what is called the second derivative of the original function:
We will see later that this enables us to write: a(t) = x''(t) , since the acceleration a of an object is equal to the time-derivative of its velocity, i.e. a(t) = v'(t) .
It can be shown, from the above definition for the derivative, that derivatives satisfy certain properties:
- (P1) (f + g)' = f' + g'
- (P2) (cf )' = cf' , where c is a constant.
- (F1) if f (t) = t n , where n is a non-zero integer, then f'(t) = nt n-1 .
- (F2) if f (t) = c , where c is a constant, then f'(t) = 0 .
- (F3a)if f (t) = cos wt , where w is a constant, then f'(t) = - w sin wt .
- (F3b)if f (t) = sin wt , then f'(t) = w cos wt .
Velocities Corresponding to Sample Position Functions
Since we know that v(t) = x'(t) , we can now use our new knowledge of derivatives to compute the velocities for some basic position functions:
- for x(t) = c , c a constant, v(t) = 0 (using (F2))
- for x(t) = at 2 + vt + c , v(t) = at + v (using (F1),(F2),(P1), and (P2))
- for x(t) = cos wt , v(t) = - w sin wt (using (F3a))
- for x(t) = vt + c , v(t) = v (using (F1),(P2))
Acceleration in One Dimension
Just as velocity is given by the change in position per unit time, acceleration is defined as the change in velocity per unit time, and is hence usually given in units such as m/s 2 (meters per second 2 ; do not be bothered by what a second 2 is, since these units are to be interpreted as (m/s)/s--i.e. units of velocity per second.) From our past experience with the velocity function, we can now immediately write by analogy: a(t) = v'(t) , where a is the acceleration function and v is the velocity function. Recalling that v , in turn, is the time derivative of the position function x , we find that a(t) = x''(t) .
To compute the acceleration functions corresponding to different velocity or position functions, we repeat the same process illustrated above for finding velocity. For instance, in the case
we find a(t) = v'(t) = a ! (This suggests some method to the seeming arbitrariness of writing the coefficient of t 2 in the equation for x(t) as a .)
Relating Position, Velocity, and Acceleration
Combining this latest result with (2) above, we discover that, for constant acceleration a , initial velocity v 0 , and initial position x 0 ,
This position function represents motion at constant acceleration, and is an example of how we can use knowledge of acceleration and velocity to reconstruct the original position function. Hence the relationship between position, velocity, and acceleration goes both ways: not only can you find velocity and acceleration from the position function x(t) , but x(t) can be reconstructed if v(t) and a(t) are known. (Notice that in this particular case, velocity is not constant: v(t) = at + v 0 , and so v = v 0 only at t = 0 .)
A natural question to ask might be, "Why stop at acceleration? If v(t) = x'(t) , and a(t) = x''(t) , why don't we discuss x'''(t) and so forth?" It turns out, the third time derivative of position, x'''(t) , does have a name: it is called the "jerk" (honestly). The nice thing is, however, that these higher derivatives don't seem to come into play in formulating physical laws. They exist and we can compute them, but when it comes to writing down force laws (such as Newton's Laws) which deal with the dynamics of physical systems, they get completely left out. This is why we don't care so much about giving them special names and computing them explicitly.