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Portfolio Problem #1
By Michael Gehlke
October 25, 2004
We are given:
“Bret Boone of the Seattle
Mariners hits a baseball one night at an angle of 28° with an initial velocity
of 126 feet per second. He hits the
ball 3.2 feet above the ground and sends it toward the part of the outfield
where the fence is 15 feet high and 380 feet from home plate.”
And the following problems:
1.
Find the parametric equations that can be used to model Bret
Boone’s hit. Simplify your equations as
much as possible.
2.
Draw a graph of the flight of the ball. Indicate direction of travel.
3.
Determine analytically whether or not the ball will clear then
fence.
4.
Determine analytically when the ball will hit the ground and
how far away from home plate assuming that there were no obstacles.
5.
Fine the value of dx/dt .
Explain the significance of this derivative in terms of the given
context.
6.
Find the value of dy/dt . Explain the significance of this
derivative in terms of the given context.
7.
Determine analytically the maximum height using a derivative.
8.
Use derivatives to determine analytically the speed of the
ball at t=0.5 and t=2.5 seconds.
9.
Find the value of d2x/dt2. Explain the significance of this derivative
in terms of the given problem.
10.
Find the value of d2y/dt2. Explain the significance of this derivative
in terms of the given problem.
11.
Explain how you can quantify the acceleration of the ball in
this problem at t=0.5 and t=2.5 seconds.
12.
Is the ball speeding up or slowing down at these moments in
time. Justify your answer with mathematical
evidence.
13.
In what ways is analyzing motion of parametric models similar
to analyzing motion of simple horizontal and vertical models? In what ways is it different?
Part One:
Find the parametric equations that can be used to model Bret Boone’s hit. Simplify your equations as much as possible.
Information:
First, let us take a look at the base projectile equations
we are given:
Where:
A is the initial velocity in feet per second, q
is the launch angle, and B and D are the initial horizontal and vertical
displacement (respectively) of the object.
Now, to take a look at the information we have. We are given that the initial velocity (A)
is 126 feet per second; that the launch angle (q) is 28°; and
that the ball was hit 3.2 feet above the ground (D).
Assumptions:
For this problem, we must assume that the ball is moving in
two dimensions, and that it was hit directly over home plate.
Solution:
To solve this problem, we simply have to come up with the
equations that will model the hit. To
do this, we must plug the information above into the equations given above, and
then simplify as much as we can.
As
we can see here, 
we simply need to
insert the values we
are given and
simplify to get
our solution.
As you can see from those, the equations that model the
movement of the ball are:
or 
(If
you solve for the sin/cos part using your calculator)
Part Two:
Draw a graph of the flight path of the ball. Indicate the direction of travel.
Information:
The information needed to solve for this problem are the two
equations used to model the motion that we found in Part One.
Assumptions:
Solution:
Using our graphing calculator, we find that the graph of the
functions looks like this:
Part Three:
Determine analytically whether or not the ball will clear the fence.
Information:
Our information for this section of the problem are the two
displacement equations that we found in part one, that the fence is 380 feet
from the start, and that the fence is 15 feet high at that point.
Assumptions:
We must assume that the ball will clear if the value is
greater then 15 feet, since the problem never gave us the size of the ball.
Solution:
This problem must be solved in two steps, the first being to
figure out the value of t for which the ball will be at the same distance as
the fence from home plate, and the second being to determine whether or not the
height of the ball at that value of t is greater or less then the threshold of
15 feet to clear the fence.
Information:
All we need for this problem is the equations that we established
in Part One, and what common sense tells us, that the ground is 0 elevation.
Assumptions: That the ground is equal to 0, that there are
no obstacles the ball will strike on its path.
Solution:
To solve this problem, all we need to do is to set the
vertical displacement function equal to 0, solve for t, and then enter the
found value of t into the horizontal displacement function.
Using the vertical displacement function:
Now, having found the times at which the vertical distance
equals to 0, we much decide which is the one to use. This is not hard since we are looking for a time after the ball
was hit, hence a positive time.
Now, we must enter this time into the horizontal
displacement function to see the distance the ball has traveled in this time.
Therefore, the ball will hit the ground about 417.2390154
feet from home plate.
Part Five:
Find the value of dx/dt . Explain the significance of this derivative in
terms of the given context.
Information:
We need the vertical displacement function found in Part
One.
Assumptions:
Solution:
This answer must be broken down into two parts. The first is the taking of the actual
derivative of the function. The second
is explaining the meaning of the derivative.
So for the first:
Now, for the second:
This derivative is important because it is the horizontal
velocity of the function, and because it is one of the components needed to
figure of the vector of the ball. It is also in this case a constant.
Part Six:
Find the value of dy/dt . Explain the significance of this derivative in
terms of the given context.
Information:
We need the vertical displacement function found in Part
One.
Assumptions:
Solution:
This answer must be broken down into two parts. The first is the taking of the actual
derivative of the function. The second
is explaining the meaning of the derivative.
So for the first:
Now, for the second:
This derivative is important because it is the vertical
velocity of the function, because it is one of the components needed to figure
of the vector of the ball, and because in this case it is the variable
velocity.
Part Seven:
Determine analytically the maximum height [of the ball] using a derivative.
Information:
The information we need for this problem is the vertical
displacement function and the derivative of it.
Assumptions:
Solution:
To determine the maximum height of the ball, we must first
locate the value for t at which the ball ceases to go upwards. This can be found by setting the derivative
of it, which tells us the instantaneous slope, to 0, since at that point, it
will neither be going up or down, hence at the highest point of its arc.
So as we can see from this equation, the time at which the
ball reaches its peak is about 1.848544278 seconds after it is hit.
We now need to enter this into our vertical displacement
equation to see what the height of the ball was at this time.
As we can see from this, the height of the ball at its peak
is about 57.87385519.
Part Eight:
Use derivatives to determine analytically the speed of the ball at
t=0.5 and t=2.5 seconds.
Information:
For this problem, we need the derivatives found in Part 5
and Part 6, along with the information stated in this question.
Assumptions:
Solution:
To determine the speed of the ball, we must take into
account the speed of both vectors, and then combine them to find the speed of
the vector the ball it on. While I’m sure
that there are many ways to do this, I have chosen to find the general equation
and then put in the values of t we were given at the start of the problem.
First off, to find the general equation, we must look at
Pythagorean theorem. 
Now, we look at this
and realize that A is our dx/dt function, and that B is our dy/dt
function. That would mean that C is the
actual vector of the ball.
Now, we can solve for the general equation.
Now that we have the general equation, we can simply enter
into our calculators the values of t that we were given.
Doing this, we find that:
At t=0.5, the speed is about 140.833579728 feet per second
and;
At t=2.5, the speed is about 135.670408767 feet per second.
Part Nine:
Find the value of d2x/dt2. Explain the significance of this derivative in terms of the given
context.
Information:
The information we need for this problem is the derivative
of the horizontal displacement function that we found in Part 5.
Assumptions:
Solution:
This problem must be divided into two portions, the first
being to take the derivative, and the second being to explain the significance.
For the first part:
This is very easily arrived at; being that constants drop
out during derivation, hence the second derivative of the horizontal
displacement function is 0.
Now, for the second part, why this derivative is important
to the problem. This derivative is
important because it is the horizontal acceleration in this equation. Because it is constant, and equal to 0, there
is no forward acceleration.
Part Ten:
Find the value of d2y/dt2. Explain the significance of this derivative in terms of the given
context.
Information:
The information we need for this problem is the derivative
of the vertical displacement function that we found in Part 6.
Assumptions:
Solution:
This problem must be divided into two portions, the first
being to take the derivative, and the second being to explain the significance.
For the first part:
This is fairly easily arrived at because it requires only
two steps, the first being to drop the constant, and the second to be
subtraction 1 from the t exponent and then multiplying the coefficient by the
original power. In this case the
original power was 1, so the t component simply dropped out of the equation.
Now, for the second part, why this derivative is important
to the problem. This derivative is
important because it is the vertical acceleration in this equation. In this case, it is the only acceleration,
and it is responsible for all the changes in speed throughout the entire
equation.
Part Eleven:
Explain how you can quantify the acceleration of the ball in this problem
at t=0.5 and t=2.5 seconds.
Information:
The information that we need for these two problems is the
information found in the previous two problems, Part Nine and Part Ten.
Assumptions:
Solution:
The solution to this problem is more analyzing data rather
then solving and producing any new figures.
We can quantify the acceleration rather easily by simply
looking at it. Since there is no
horizontal acceleration, thus all of the acceleration is in the vertical
dimension. We have already solved for
this number and we have found it to be –32, or 32 feet per second per second in
the downward direction.
Part Twelve:
Is the ball speeding up or slowing down at these moments in time [t=0.5 and
t=2.5]. Justify your answer with
mathematical evidence.
Information:
The information needed to solve this problem in the manner
that I did is the general speed equation that I found in Part Eight.
Assumptions:
Solution:
The solution to this problem requires that we first define
what it is that we are trying to find.
We are trying to find whether the ball is gaining speed or losing
it. The next task would be to determine
the best way to do this. I have decided
that observing the rate of change of speed of the ball traveling along its
vector was the best way to do this task.
Given that, the first step would be to derivate the general
speed equation that we found in Part Eight. This will give us the general rate
of change of speed equation for the ball in relation to its current vector.
Now the work comes down to entering values and interpreting
their meanings. For this, negative
values indicate that the ball is slowing down, since it will be applying force
against the current movement along the current vector. Given that, positive values will indicate
gaining speed, since it will be adding force to the current movement along the
current vector.
Having established that,
at t=0.5 we find that the value is about –9.8052562725,
hence it is slowing down, while
at t=2.5 we find that the value is about 4.91699453892,
hence the ball is speeding up.
Part Thirteen:
In what ways is analyzing motion of parametric models similar to analyzing
motion of simple horizontal and vertical models? In what ways is it different?
Information:
Only what we know about the subject matter.
Assumptions:
That we know something about the subject matter.
Solution:
Analyzing motion of parametric models is similar in the
sense that it is still in a plane, and that you can manipulate a single part of
the function in the same manner, to find say the acceleration in respect to one
of the movements.
That however is really where the similarities end. It is quite different in other respects
because you have to keep in mind that the graph that you see on the screen is
actually the result, not the relation between the variable and the output. It is also different because you have to
constantly keep in mind how the two kinds of movements are interacting with
each other.
