Download Section 1

Free-Fall: An object that is dropped falls down due to the gravitational influence of
Earth. If the object experiences no (or, at least, very little) air resistance, then it is in
"free-fall". An object that is in free-fall increases its velocity by 9.8 m/s every second
(this is equivalent to 32 feet/s every second). We therefore say that its acceleration is 9.8
m/s2. If an object is thus thrown downward initially with a velocity of, say, 5 m/s, then
after 1 second its downward velocity is 5 + 9.8 = 14.8 m/s, after two seconds its
downward velocity is 5 + 9.8 + 9.8 = 24.6 m/s, after three seconds its downward velocity
is 5 + 9.8 + 9.8 + 9.8 = 34.4 m/s and so on. In equation form, we say that v = v0 - gt
where v is the velocity of the object in free-fall after time t; v0 is the initial velocity of the
object (positive if the object is thrown upwards, negative if the object is thrown
downwards) and g represents the acceleration of the object due to gravity, 9.8 m/s2. It is
possible for an object to be moving upwards and still be in free-fall. This will occur
when the initial velocity of the object is positive, i.e., up. This also means that there is an
acceleration acting downwards even when the object is moving upwards or even if the
object has a velocity of zero. This again goes back to the situation of one dimensional
motion and having a velocity of one sign and an acceleration of an opposite sign.
The equation above is the relationship between velocity and acceleration when the
acceleration is constant. We can also find a relationship between velocity and
acceleration and displacement. We are aware that if the velocity is constant, then the
change in displacement is d = vt where d is the change in displacement, v is the velocity
and t is the time of travel. If, however, we started not at the origin but at some location d0
then our more general equation becomes d = d0 + vt. Now, though, suppose the velocity
is changing, starting with an initial velocity of v0. It turns out that the relationship
becomes d = d0 + v0t + ½at2. Note that the displacement depends upon the square of the
time. We say that this formula is "quadratic". In many cases, we are able to say that the
initial location of the object is 0, i.e., d0 = 0. Further, if an object starts from rest, then
that means that v0 = 0. Thus, for an object starting at rest, the equation for distance
traveled simplifies to d = ½at2. If the acceleration is known, as is the case for free-fall,
then we can solve this equation in terms of t to get t = (2d/a)1/2; this can arise when we
know the distance the object has traveled, but we wish to know for how long it has been
traveling.
In summary, the relevant equations are:
xf = xi + vi t + ½ at2
vf = vi + at.
A useful equation is vf2 = vi2 + 2a(xf – xi). (In these equations, the letter x essentially
plays the same role as the letter d used in the discussion above.)
Newton's First Law of Motion: Aristotle formulated the concept that the natural state
of all bodies was to be at rest--after all, if I start anything moving, then that anything will
eventually come to a rest. Unfortunately, Aristotle didn't have a good grasp of the
effects of friction. Isaac Newton did. He realized that if all external forces were removed
from an object, then the motion of that object would not change. This was Newton's first
law of motion.
In our study of motion, we need to start clarifying the concept of mass. In effect, we will
find that mass is a measure of how hard it is to get an object moving. Applying a force to
a light object will get the object into significant motion. Applying that same force to a
heavy object will only seem to barely influence that object into motion. Also, in
everyday-speak, we use the words "mass" and "weight" interchangeably. However, in
physics-speak, they are two different quantities. "Weight" is dependent upon the gravity
in which the object exists. For instance, since the Moon is lighter, the gravitational
acceleration felt on the surface of the Moon is less than the gravitational acceleration felt
on the surface of the Earth. Therefore, an object on the Moon weighs less than an object
on the Earth. However, no matter where it is, that object has the same mass. Consider a
baseball and a battleship in outer space where they would be weightless. If a given force
is applied to the baseball, significant motion can be gained by the baseball. However, if
that same force were applied to the battleship, the motion of the battleship would be only
barely affected. That's because the mass of the battleship is so much more than the mass
of the baseball. We can also say that the battleship has more inertia than the baseball.
Newton's Second Law of Motion: You've likely noticed by this point in your life that
not all motion acts in a straight line. From Newton's First Law, we know that if no forces
acted on a moving object, it would move in a straight line. Therefore, if there is nonstraight motion, then there must be a force. Newton's Second Law gives us a chance to
start quantifying the force and its effects on the moving object. Newton's Second Law
states that a non-zero force will impart a non-zero acceleration to a mass. In fact, the
acceleration that the mass gains is proportional to the force acting on it--if the force
acting on the mass doubles, the acceleration of the mass doubles and if the force triples,
the acceleration triples. The direction of the acceleration is also in the direction of the
applied force. Also, the acceleration gained by the object is inversely proportional to the
mass of the object, so if the mass of the object doubles, the acceleration is cut by one-half
and if the mass of the object triples, the acceleration is cut in magnitude by one-third. In
equation form, we say a = (Fnet/m) where the boldface notation indicates vector quantities
and lets us know the acceleration is in the direction of the net force on the object.
What do we mean exactly by the "net force" or Fnet? Suppose two forces act on the mass
(honestly, this happens all the time--frequently, there are quite a few more than two
different forces). We can represent these forces as arrows with lengths equal to the
magnitude of each force, emanating from the object and each pointing in the direction of
each force; in the most general situation, they won't be at right angles to each other. The
net force is the vector sum of the two individual forces. We can show this graphically by
considering each of the two vectors as the sides of a parallelogram with one vertex where
the object is located. We can then complete the other two sides of the parallelogram and
draw a diagonal from the vertex where the object is to the opposite vertex. This diagonal
represents the net force acting on the object due to the two individual forces (see the
diagram done in class for the graphical portion of this discussion). There is a mathema-
tical way to come up with the exact value of this net force; this involves trigonometry and
won't be discussed in this course, though.