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: http://zebu.uoregon.edu/1998/ph101/l13.html
Дата изменения: Fri Nov 13 23:05:32 1998 Дата индексирования: Tue Oct 2 01:03:44 2012 Кодировка: Поисковые слова: sculptor galaxy |
We will be studying projectile motion both in the atrium and in Applet land. For now, the following two graphs summarize the kinds of observations you can make about projectile motion:
Relevant parameters:
Out task now is to see how these parameters can be predicted by knowing the initial velocity and angle of a projectile.
Let's review the data that we acquired in the atrium:
Some questions:
Determining the X and Y components is fairly simple as shown in the
above diagram. Any velocity vector can be deconvolved into
its X and Y components by using sine and cosine functions on the
angle, A.
For instance, the the angle A was ninety degrees then there would be
no X-component to the velocity. Since the cosine of 90 degrees is
0, then Vx cosine 90 = 0 and there is no X component to
the velocity.
Similarly, if the angle is 0, then
Vy cosine 0 = 0 and there would be no Y component to
the velocity. Any other angle would have both X and Y components
as defined in the figure above and again below:
2. Why does the maximum distance occur at 45 degrees?
One of the students in class supplied the
correct answer by staying that this is the angle in which the
product Vx * Vy is a maximum.
This conclusion was reached by understanding, from the atrium
experiment, that both the X and Y components of the velocity
are important.
How did we know this? We knew this from the 30 vs 60 degree
experiment. We established that Vx in the 30 degree
case was larger than Vx in the 60 degree case.
Because the 60 degree case achieved a higher maximum height, we
also established that Vy was higher in the 60 degree
case than the 30 degree case.
But, both cases produced the same horizontal range which indicates
that both Vx and Vy are important in
determining the total range.
The reason that 45 degrees gives the largest horizontal range
is because the product:
Another student correctly pointed out that this is merely a consequence that cosine 30 = sine 60 and sine 30 = cosine 60 and so the two cases are symmetric.
4. How can we determine the total time of flight?
If you shoot the cannon below you
will see a parabolic trajectory. Defining the motion in the
two coordinates allows us to say the following:
In the X-coordinate the shot starts from the cannon, call this
xo and then hits the ground at some other
distance, call this xf .
The total time it takes for the projectile to cover this distance
in X only depends on the Velocity in the X-direction. Hence the
time of flight can be expressed as:
Now let's consider just the Y-motion. In the Y-direction, the
projectile reaches some maximum height, h, and then returns to the
ground. So the time it takes to undergo this motion is
Now note the following: Vy, unlike Vx, is not a constant, but is continually changing. When the object is at maximum height, its velocity is, of course Vy = 0. So there is some average velocity:
1/2(Vy) that holds for the motion of the projective from height = 0 to height = h. Since the object has to go up and down, its total displacement is 2h but its average velocity during this total displacement is 1/2(Vy).
So the travel time t is
Well, the students correctly inferred that the maximum height depends on the initial velocity in the y-direction as well as inversely on the gravitational acceleration, g
So that leads to something like:
h = Vy/g
But if we look at the units of that we see that its:
meters = (meters/sec) / (meters/sec2)
but the right had side has units of seconds. Hence, dimensionally we need to multiply the right hand side by meters/second to get meters. This means that we must have something like:
h = Vy/g * V
which would now be dimensionally correct. But what's the physics for this?
Next we did an in class demo about free fall and the measurement of velocities. That lead us to the general result that free fall represents a case of gravitational potential energy converted into kinetic energy. This is also the point of the second lab activity.
Since energy is mostly conserved in free fall (little energy is lost due to air friction). We can write down the conservation of energy between potential energy and kinetic energy or:
or
v2 = 2gh
or
h = v2/2g
We see that the mass drops out entirely which, of course, is why
two objects of different mass fall at the same rate. If heavier masses
hit the ground prior to light masses when dropped at the same time,
there would be a violation of energy conservation.
So what determines the maximum height is the initial kinetic energy
of the projectile and the angle of launch.
6. What determines the total range?
Now we are in an position to understand what the total range of the projectile is:
Return to the earlier time of flight analysis where we established that:
But we just derived the expression h = Vy2/2g
We can then substitute this expression into the time of flight equality to get:
Letting R = range = xf - xo then leads us to the final expression:
or
R = (2V2cosine A sine A)/g
Using the cannon below you can verify this expression directly by measuring the distance using the cursor readout feature now enabled.
To summarize, we were able to derive the correct expression for range of the projectile by using a time of flight analysis in combination with convervation of energy.
7. How does energy conservation help to determine total range and maximum height?
Consider this as a question for the final exam.
The cannon will help: