Constant acceleration in one dimension
A very important special case of motion of a particle is the motion
of a particle with constant acceleration along the x-axis.
If the acceleration is constant then the slope of the tangent line to
the vx versus t curve is constant. This can only pertain to a linear vx
versus t graph,
[c.f, y(x) = mx + b]. Using the definition of
ax = dvx/dt , multiplying this by dt and integrating gives,
vx(t) = vox + ax t
(1d, ax=constant).
In this equation, vox is the velocity of the particle when the
stopwatch reads zero, "ax" is the constant acceleration of the
particle, "t" is the time, and vx(t) is the velocity at time t.
As will be shown in class using calculus, x(t) - xo , is the area
under the vx(t) versus t curve between time zero and time, t.
Therefore, using vx = dx/dt, multiplying by dt and then integrating
gives,
x(t) = xo + vox t + 1/2 ax t2
(1d, a=constant).
(Note: memorize this only, since dx/dt gives the vx(t) eq. .)
In the equation, xo is the position of the particle when the
stopwatch reads zero, and x(t) is the position at time "t".
Any constant acceleration problem may be solved with ONLY the
above two equations, however, sometimes time can be saved by
using the "shortcut" equation (eliminating "t" from the above two
equations):
vx2 = vox2 + 2 ax [x -xo]
(1d, ax=constant, shortcut).
Proof provided below by the reader
Free fall is a special case of the above with ax = -g (x-axis pointing
up away from earth) or ax = g (x-axis pointing down). Near the
surface of earth and neglecting air resistance we have g = 9.8 m/s2.
Philosophical aside:
The above equations are extremely powerful. Given the initial
conditions, xo and vox, we are enabled to predict the future (and
past) mechanical state (x and vx) of the system. In philosophy, we
say the system is deterministic because there is no room for "free
will". If humans obey deterministic mechanical laws then our
actions and thoughts are fully determined and therefore, we are not
morally responsible for our actions and thoughts. However,
modern physics seems to suggest we are not fully determined but
you will have to study modern physics (i.e. third semester
introductory physics) to understand how this comes about.
[examples: in class]