Rodney Garland
PLAB 223 – 01
10/16/11
Lab #5 – Projectile Motion
Introduction -
This lab entertained the idea of projectile motion and how, at different maximum heights and velocities, an object can fly shorter or farther distances. The point of the lab was to find the initial velocity of the projectile launched, as well as the final distance it reached. Topics and ideas that were key to this experiment include: normal force, air resistance, the work energy principle, the momentum priciple, and the idea of uncertainty.
Procedure -
The first step was to set up the launcher at an angle of 0 degrees. A ball of mass m was then loaded into the cannon and pushed back until one click was heard. The reason behind stopping at one click, is that two or more clicks may add too much power to the shot, sending the ball on a dangerous path of destruction. After prepping the launcher, two meter sticks were obtained and laid flat on the ground, end to end. The meter sticks were then kept in place by use of duct tape. Once the meter sticks were locked into place, two sheets of paper were placed on the ground in places where the ball was thought to land after launch. Finally, the pieces of paper were overlayed by pieces of carbon paper to allow the shots to be measured after impact. The second part of the experiment was just a repeat of the first, except the launcher was set at an angle of 30 degrees.
Data Collection -
Constants –
Initial Height – 1.03m +/- .002m
Mass of Ball – m
Gravity – 9.81 m/s^2
| Experiment #1 (Angle at 0 degrees) | Experiment #2 (Angle at 30 degrees) |
Shot # Distance Shot # Distance
| 1 | 183cm(1.83m) | 1 | 258cm(2.58m) |
| 2 | 188cm(1.88m) | 2 | 267.5cm(2.675m) |
| 3 | 184cm(1.84m) | 3 | 260.5cm(2.605m) |
| 4 | 190.6cm(1.906m) | 4 | 261cm(2.61m) |
| 5 | 198cm(1.98m) | 5 | 251cm(2.51m) |
| 6 | 186cm(1.86m) | 6 | 253cm(2.53m) |
| 7 | 188.3cm(1.883m) | 7 | 261cm(2.61m) |
| 8 | 192.7cm(1.927m) | 8 | 266cm(2.66m) |
| 9 | 188cm(1.88m) | 9 | 264cm(2.64m) |
| 10 | 197cm(1.97m) | 10 | 254cm(2.54m) |
| Average Distance: | 189.56cm | Average Distance: | 259.6cm |
Variance: 15cm Variance: 16.5cm
Data Modeling -
As previously stated, the point of the lab was to find the initial velocity of the projectile after launch. Once the data was collected, either the work energy principle or the momentum principle can be used to figure this out.
The Work Energy Approach:
So, what do we know? Well, we know that the work energy principle states that,
W = Delta E
W = mgh; Delta E = 1/2m(vf)^2 – 1/2m(vi)^2
And so, mgh = 1/2m(vf)^2 with the height starting at 1.03cm
With this in mind, we also know that,
m = mass of the ball
g = 9.81 m/s^2
h = 1.03cm
And now, the only thing left to find would be our velocity.
The Momentum Approach:
If the work energy principle isn't working out, this would be your other option. The question here is the same as before, "What do we know?" Well, we know our kinematic equations.
xf = xi + vx(t) (Assuming time at 0)
yf = yi + vyi(t) – 1/2g(t)^2
vfy^2 = viy^2 – 2a(Delta y)
vfx^2 = vix^2 – 2a(Delta x)
In this example, we have these variables on hand.
yi = 1.03cm (The initial height of the launch)
yf = 0 (The ground)
g = 9.81 m/s^2
To find the initial velocity for the horizontal direction, one would simply have to measure the final velocity after launch. Once that number is obtained, they would have to plug it into the formula and then they will have found the initial velocity. The problem with this approach is that it adds more data collection to the experiment, finding the final velocity. However, it is still possible, it just involves a little bit more work.
These two approaches will work when dealing with both experiment #1 and experiment #2. The only difference is the change in the angle the ball was launched. This change in angle affects the velocity in such a way that,
vxi = vx(cos(Angle))
And, vyi = vy(sin(Angle))
Conclusion
In conclusion, an experiment like this raises many questions. A few of those questions I will attempt to answer are, "Does the mass of the ball affect the distance the ball will travel?" and "How does uncertaintly come into play when dealing with this experiment?"
To answer the first question, I would like to start by bringing up the idea of air resistance. Although, if you apply the work energy principle to solve for your initial velocity, you'll see that m, which in this case represents the mass of the ball, is part of the equation. However, if you look even further into this question, one could certainly entertain the idea of air resistance to further explain the situation.
To begin, the formula for air resistance is,
F(air) = 1/2pAcv^2
Where,
F(air) = mass of the object x acceleration
p = The density of the fluid (air)
A = The area of the object
c = The drag co-efficient
v = The magnitude of the velocity
Thanks to this formula, it can be shown that a change in the mass of an object will effect the final distance the projectile will reach, thanks to air resistance. However, if the mass isn't that different, for instance, a difference of .00005, the effect of said mass would be inconsequential to the final measurement of the projectile's distance.
The second question can be answered very quickly and simply. In this experiment, uncertainty can be represented in both human error and rounding decimal places. The human error could be found by the ways the measurements were taken. Either a misread of a measurement tool or misuse of a mathmatical formula can be attributed to human error. In this case, the variables that contain uncertainty due to human error include: the initial height and the variance on the final distance calculations. As for decimal places causing uncertainty, a bunch of rounded numbers could cause a loss of precision, if the numbers were rounded prematurely. A quick example will show that,
20.567 + 40.678 = 61.425
And, 21 + 41 = 62
As the list of numbers grow, the final product could wind up a lot different than if a few more decimal places were carried into the equation. A loss of precision like that, could cause rocket ships to explode, which would be a very bad thing.
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