L05.  Range of a Projectile

About your lab report:  In addition to the starred items given in the lab guide, your report for this lab must include the following sections:  Theory, Apparatus, Method, Legend of Symbols (if new symbols are introduced), Error Analysis (quantitative), Error Analysis (qualitative).  A graphical analysis is not required.  More details are provided below.  You'll submit your report as a word-processed document.

About working with a partner:  You may work with one partner on portions of this lab. If you do so, pay close attention to the following guidelines for cooperative and independent work.

1. You must write the Theory and Prediction section on your own.
2. You may work with a partner in designing and setting up the experiment.
3. Each person must take their own set of measurements, do their own calculations, and test their results.
4. You must write your report independently of other students. Do not work together on report writing. Your writing must be your own.

About preparing for lab day: Read these instructions completely so that you'll know what you'll be doing on lab day. Submit the Theory and Prediction section prior to the lab according to instructions from the teacher. Prepare your two data tables in advance.

Goal

Predict the range of a horizontally-launched projectile and test the result experimentally.

Introduction

This lab exercise will give you real-world experience with the physics of projectiles.  Here's an overview of the method.

A ball rolls down a ramp and then across a level table as shown below.  The ball is stopped before rolling off the table.  The average speed of the ball on the table is measured by finding the time for the ball to travel between two markers (labeled d_i and d_f in the diagram.)

Knowing the speed, the equations of projectile motion are used to predict how far horizontally beyond the edge of the table the ball will be projected.  The stop is removed and the ball is released on the ramp in the same way as before in order to see how close the ball comes to the predicted range.

Prelab: Theory and Prediction

The theory you work out below must be submitted in advance of the lab. For your final report, include this work in the Theory section with any necessary corrections.

In any experimental design, the experimenter needs to know the goal and the tools available to do the experiment and take measurements.  In a physics lab, the experimenter also needs to know some physics in order to guide the measurements.  You already have experience with the physics of projectiles.  You need to decide what to measure in order to predict the range of the ball beyond the edge of the table.  We're pretending for the time being that you can't measure the range directly by placing a meter stick on the floor.

Follow the guidelines for solving 2-dimensional dvat problems to work out the physics in order to obtain an equation for the range--call it R--in terms of the launch velocity, v_ox, and the vertical displacement, y - y_o. Avoid having quantities in your equation that you can't measure accurately or easily. Such a quantity, for example, would be the time for the ball to fall to the floor.  While you'll include that time in the equations that you write down first, you'll need to eliminate it from the final equation.

Equipment and setup

You'll need the following items or substitutes that work in the same way.  Use your ingenuity in setting things up.

• A table about 4-ft long or longer
• 5 or 6 books or comparable substitute
• A ball an inch or less in diameter.  The ball has to roll smoothly.  Glass marbles work well.
• A piece of flexible cardboard or plastic that you can bend into a ramp.  The ramp has to curve so as to gently release the ball onto the table without much of a bounce.
• A stack of books to hold the ramp in place at the upper end.  You can bend a flap into the cardboard at the upper end and slip it between two books to hold it in place.  (See diagram to the right.)
• Sticky tape (such as masking, duct, Scotch) to stick the lower end of the ramp to the table and also place markers at d_i and d_f.
• A meter stick or tape measure.  You could make do with a 30-cm ruler, but you'll have to place it end-for-end for longer distances.
• A stopwatch. You need to be able to measure time to tenths of a second or better.  Alternatively, you may use a photogate (standalone or interfaced to a computer). In the latter case, you may need assistance from a science teacher.

Method

Different students may be using different items of equipment and have differences in their methods.  Therefore, it's important for this lab that you include Apparatus and Method sections in your lab report. The latter will include not only the method of timing the ball, but also your method of determining R.  Be sure to include a diagram.  If you use a photogate, include a description and a diagram of how you positioned the photogate for best results.

If you are working with a partner, each partner must obtain an original set of data. Students with the same data will receive no credit for their data or calculations.

Instructions intended for partners are given in bold, red font.

Data:  Times and Distances

One partner takes a set of data as described below.

Make a change to the height of the ramp but be sure to do in such a way that you can put the ramp back into its original position later. The partner who hasn't taken data will use this new height and repeat the instructions in the table above.

Now do the calculations for your set of times. Calculate the mean (simple average of the 5).  Calculate the deviations as follows:

Deviation = | Time Trial - Time Mean |

That is, find the absolute value of the difference between each time trial and the mean of all the trials. 

Now calculate the mean of the deviations.  The mean deviation is a measure of the reproducibility of your measurement technique.  Another measure that is more typically used in statistics is the standard deviation.  This is the square root of the sum of the squares of the deviations.  You probably have a function on your calculator for that.  For this lab, we'll stick with a simple average of the deviations.

It's useful to express the percentage that the mean deviation is of the mean.  That gives you a better idea whether it's a significant source of error.  Go ahead and calculate percentage mean deviation as follows:

% Mean Deviation = 100 (Mean Deviation) ÷ Mean

Enter your result in the table.

In all your calculations, be sure to pay attention to significant figures.

In the theory, you listed those things that you need to measure in order to calculate the range.  Measure and record them now.  All measurements must be included in your data section. If you made any distance measurements in feet, convert them to meters.

Substitute your measurements in SI units into your formula for R and calculate the value.

Testing your prediction

Since the ramp will still be set up for the last partner who measured time trials, the following test should be carried out by that person first.

Now restore the ramp to the height used by the partner who measured time trials first. Then carry out the instructions in the table above.

Error Analysis

You may have noticed a significant difference between your calculated and measured ranges (or maybe you didn't).  This may be a result of inaccuracy in your timing technique, or it may be a result of some other factor or combination of factors.

You may have also noticed that the percentage mean deviation for your range measurements was less than that for your time measurements (or maybe you didn't).  The latter would indicate that your technique for releasing the ball on the ramp was more consistent than your timing technique.  That wouldn't be surprising.  Releasing the ball only requires holding it at a particular location and letting go.  Timing requires starting and stopping the stopwatch while watching the ball move rapidly past two marks.

There's a simple method to estimate the total percentage uncertainty in your prediction of range.  Simply add the percentage uncertainties in the quantities that you use to calculate the range.  For time, you could use the percentage mean deviation as the measurement uncertainty.  For distance, just estimate your uncertainty in measuring a particular distance and then find what percentage that is of the distance measured. 

For example, suppose you measured the distance d_f - d_i to be 0.841 m, and you estimated that you could be off by a millimeter at each end of the measurement.  The percentage uncertainty would be 0.002 m ∙ 100/0.841 m = 0.2%.  Note that the result is rounded to 1 significant figure, since 0.002 has 1 significant figure.  Note also how small 0.2% is compared to your percentage uncertainty in time.  This tells you that the only important source of measurement error in the experiment is timing.  Suppose, for example, that your timing uncertainty is 5%.  When you take significant figures into account, the sum of 5% and 0.2% is still 5%.  While you also need to add into that the percentage uncertainty in measuring the height of the table, that also will be small in comparison to the uncertainty in time.

Can you say that the difference between your calculated and measured ranges was the result of your timing uncertainty?  You can't be sure, but it's a good bet that the timing uncertainty is a significant factor.  Is it the only factor?  Calculate the percentage difference between your calculated and measured ranges. Note that this process is used to compare two values when you don't have a strong reason to expect that one value is better than the other. Note also that the sign of the result tells you by inspection which value is the larger. This is helpful when looking for systematic errors in a measurement technique.

% Difference = 100 ∙ (calculated range - measured range) ÷ (calculated range + measured range)

If the percentage difference is significantly more than your timing uncertainty, then there are most likely other factors that contribute to error.  Let's try to identify them.  We want to look for large errors rather than those tiny 0.2% errors in distance measurements.  We also want to be specific in identifications.  The phrase 'human error' is non-specific and should never be invoked as a source of error.  On the other hand, here's something specific:  Suppose your timing technique were such that you consistently started the stopwatch too soon and stopped it too late.  Then your times would be consistently large.  Due to their consistency, that wouldn't contribute to a large mean deviation in your time measurements.  You could have a small deviation and still have a large error.  Such an error is called systematic, because it results from your system of measurement.  It's very difficult to know whether you're making such an error.  Sometimes watching how other people measure can help you see what you may be doing differently.

In the case of the systematic timing error described above, the measured time of travel of ball across the table would be larger than actual.  That would make the calculated speed of the ball smaller than actual.  Likewise, the calculated range would be smaller than actual.  So if you saw, in fact, that the calculated range was too small, you could say that one possible reason was the systematic timing error mentioned above.  However, there are also possible reasons.  In your qualitative error analysis, discuss two other reasons that could explain the calculated range being smaller (or perhaps larger) than the actual range.

More about your lab report

This report requires both quantitative and qualitative error analysis.  Use the section above as well as this sample report to guide you.  Prepare your lab report as a single word-processed file. You can use the Word drawing tools for your diagrams or, if you prefer, you can draw diagrams in a different program and paste them into your document. If you don't want to use a drawing program, you can draw diagrams neatly by hand. (The same applies to writing equations.)  In order to do so, leave space in your word-processed document for diagrams and equations. Then print out your document and add the missing items by hand. If you use this latter method, you'll need to submit your report as a paper copy.

If you worked with a partner in setting up the experiment and taking data, write the partner's name in the upper left-hand corner of the first page of your lab report.

Submitting Your Work

If submitting your file electronically, name it in the usual way and upload it as instructed. Otherwise, submit your paper copy in the usual way.