Lab #1 - Remastered Lab Report.

Rodney Garland

PLAB 223 – 01

11/6/11

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, 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 -

Before this section begins, the following VPython Program can help demonstrate the above data in action and help explain the experiment better:

http://www.megaupload.com/?d=C8CLXSYH

As previously stated, the point of the lab was to find the initial velocity of the projectile after launch. Once the data was collected, the momentum principle can be used to solve for the initial velocity.

The question to ask is, "What do we know?" Well, we know our kinematic equations.

xf = xi + vx(deltaT) (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 formulas and then they will have found the initial velocity.

An example of this is as follows,

ax = 0m/s^2

ay = -9.81m/s^2

xi = 0m

xf = ? (This is what we're looking for.)

yi = The height of the launcher

yf = 0m (Th ground)

xf = xi + vx(deltaT)

yf = yi + vyi(t) – 1/2g(t)^2

Now, we plug in our above values and solve for t(time),

0 = yi + vyi(t) – 1/2(-9.81m/s^2)(t)^2

Once this equation is set up properly, we can use the quadratic equation to finish solving for t. Then, we use the positive value as opposed to the negative and plug that value into the first formula, as the value for deltaT. Once that's done, we can find the final distance the projectile will travel once fired.

This approach 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))

Knowing the above fomulas can also help you find at which angle the shot was made. If you were to find the initial velocity of the launch, the height of the launcher, and the final distance, you could then use the above formulas to solve this problem.

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.

Another way to solve a possible loss of precision in calculations can be the use of computers. Computers can be used to eliminate human error among other things. Using the Monte Carlo method given in class, you could look at a distribution of data related to finding things such as initial velocity in an experiment like this. In this distribution of data, you could see the measure of uncertainty from 10 shots to 100 shots. Using a computer to find mathematical solutions can help speed up the process of gathering data over time. This method of problem solving will save a good deal of time when discovering the measure of uncertainty as well as general calculations.