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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:

f'(t) =

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:

f''(t) =

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.
Without going into more detail about the mathematical nature of derivatives, we will use the following results for the derivatives of some particular functions--given to us courtesy of basic calculus.
  • (F1) if f (t) = tn, where n is a non-zero integer, then f'(t) = ntn-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.
These rules, together with (P1) and (P2) above, will give us all the necessary tools to solve many interesting kinematics problems.

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) = at2 + 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))
Notice that in this last case, the velocity is constant and equal to the coefficient of t in the original position function! (4) is popularly known as "distance equals rate × time."

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/s2 (meters per second2; do not be bothered by what a second2 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

x(t) = at2 + vt + c,   v(t) = at + v,

we find a(t) = v'(t) = a! (This suggests some method to the seeming arbitrariness of writing the coefficient of t2 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 v0, and initial position x0,

x(t) = at2 + v0t + x0

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 + v0, and so v = v0 only at t = 0.)

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