The paradigm for this unit is the ball bearing
rolling down a ramp. As the ball bearing travels down the ramp, it gains
more position in equal intervals of time (it speeds up). We found that the
velocity of the ball bearing changed by equal amounts in equal time
intervals. Upon examination
of the motion of the ball bearing, we defined it as constant acceleration.
Average Acceleration (a bar) is
defined as the change in velocity over the change in time. It is
a vector and is the slope of the v-t graph. A shorthand version of
this definition is a = v/t.
Acceleration is a rate of change of a rate
of change. An object that accelerates at 9.8 m/s^{2 }or m/s/s^{
}gains
9.8 m/s of velocity for every second it accelerates.
If the pos-t graph is a curve, the object is
accelerating. Curves that we see (to describe motion in this class) usually fit a power
regression (to the second power) or a quadratic regression. Each term in the regression
equation has some physical meaning. The shape of these curves is parabolic.
The form of the model for an object undergoing constant acceleration is y=Ax^{2}+Bx+C.
If the pos-t graph is a curve,
the slope of a tangent line to the curve is the instantaneous velocity
at that point on the graph (the velocity at that moment).
The intercept of the pos-t graph is once again the
initial position.
When an object undergoing constant acceleration
changes direction, its instantaneous velocity is zero. However, its
acceleration is not zero.
If the v-t graph is a curve, the acceleration
is changing. The slope of the tangent line is the instantaneous
acceleration for this graph.
A positive acceleration makes something traveling
in the positive direction speed up, and an object traveling in the negative
direction slows down. A negative acceleration makes something traveling
in the negative direction speed up, and an object traveling in the positive
direction slows down.
Deceleration is when an object is slowing down
(speed is going toward zero). It is not the same thing as a negative
acceleration (which can represent speeding up in a negative direction).
Working from the definitions above, we can
derive useful equations that describe the motion of objects.
In a vacuum, all falling objects accelerate
at the same rate. The shorthand for this rate is g (g on the
earth
= 9.8 m/s^{2}).
Objects accelerate at the same rate on earth
b/c even though more massive objects experience a greater pull of earth's
gravity (or weight), their mass (or inertia) resists that pull. Air
resistance interferes with this fact in the atmosphere.
The effect of air resistance is more important
as objects have the following characteristics: lighter, more surface area,
less aerodynamic shape, moving faster. We will usually ignore air
resistance b/c it makes the math easier, but for more accurate results,
it must be included in many cases.
An object undergoes freefall when only the
force of earth's gravity is acting on it in a significant way. Anything
thrown up or down, or dropped qualifies as freefalling from the instant it is
released.
The instantaneous velocity of an object tossed up
is zero at the top of its path. Its acceleration is g.
The speed at equal altitudes of an object tossed up
is equal. The velocities are opposites (one is traveling up, the other is
down).