Physics 1A
Lecture 3A
"Mathematics is the door and the key
to the sciences.”
-- Roger Bacon
Falling Bodies
Galileo was the first to predict that falling bodies
on Earth would fall with a distance proportional to
time squared (D∝t2).
Galileo surmised through repeated experiment that:
“at a given location on the Earth and in the
absence of air resistance, all objects fall with the
same uniform acceleration.”
Galileo felt that earlier work neglected the effect
that air friction (i.e. resistance) had on different
objects.
The uniform acceleration Galileo talked about was
the acceleration due to gravity, g.
Falling Bodies
In SI units, g is generally regarded as being
9.80m/s2.
g varies slightly depending on where you are on
Earth, but it always points downward towards the
Earth.
Falling Bodies
Example
In the original Superman comics, Superman
was unable to fly but could simply “leap tall
buildings in a single bound.” Superman’s
range was about one eighth of a mile (201m).
What initial velocity would Superman need to
reach his maximum height?
Answer
First, you must define a coordinate system.
Let’s choose the upward direction as positive,
and x = 0 where Superman starts at t = 0.
Falling Bodies
Answer
Let’s list the quantities we know:
Δx = +201m
a = -9.80m/s2 = constant
v=0
<-- velocity at max height
vo
<-- finding
t
<-- don’t know
•
Looks like it is the fourth equation for us:
Answer
Falling Bodies
But we know that v = 0 at max height, so it
becomes:
This is the initial velocity that Superman needs to
attain maximum height.
In class Question:
A tennis player on serve tosses a ball straight up.
When the ball is at its maximum height, the ball’s:
A) velocity is non-zero and acceleration is zero.
B) velocity is zero and acceleration is non-zero.
C) velocity is zero and acceleration is zero.
D) velocity is non-zero and acceleration is non-zero.
Vectors
Vectors are quantities that have both
magnitude and direction:
What are some examples of vector quantities?
Examples are: velocity, force, momentum.
Scalars are quantities that only have
magnitude:
What are some examples of scalar quantities?
Examples are: time, mass, energy.
Vectors
When handwritten the vector has an arrow
over it.
In a book it will be usually written in bold, as well.
The magnitude of the vector is usually written either
in italics or with the vector in the middle of absolute
value lines.
Vectors
When adding two vectors:
Make sure you take their directions into
account.
The units must be the same.
You can either perform the addition
geometrically or algebraically.
When two vectors are added together geometrically,
their vector sum is called the resultant. Vectors are
added head-to-tail.
Vectors
It does not matter what order you add the
vectors in.
When adding
multiple vectors,
just keep
repeating the
process until you
have included all
of the vectors.
Vectors
When you multiply a vector by a scalar, you
change its magnitude.
If the scalar is negative, it flips the
direction of the vector.
When subtracting vectors geometrically,
merely flip the vector with the negative
sign in front and add.
Vectors
When adding vectors, it is
usually better to break
them up into components.
This is also called resolving
the vector into components.
Vectors are usually
resolved into
perpendicular x and y
components.
When changing components
back to a resultant vector,
use the following:
Vectors
Once you have broken the vectors into
components it becomes very easy to add like
components.
Consider two vectors: A and B. First break
them up into components.
Then add like directions (x with x, and y with y).
2-D Variables
Now, as we attempt to describe 2-dimensional
motion, we need to redefine our variables.
Position will now be given by:
Displacement will be given
by:
Note that position depends
on where you choose the
origin while displacement
does not.
2-D Variables
Average velocity is again the total displacement
over the time interval:
Since Δt is a scalar and always positive,
average velocity will always point in the
direction of displacement.
Instantaneous velocity is again the velocity of
an object at any instant of time.
2-D Variables
The velocity vector will be tangent to the path
of the object.
For, example, consider the following motion:
Note that
the velocity
vector is
drawn at the
particle’s
position and
not from the
origin.
For Next Time (FNT)
Start the Homework for Chapter 3
Finish reading Chapter 3