Motion in Two Dimensions

William M Jones

 

ABSTRACT

 

            The motion in two dimensions lab experiment attempts to determine the dependence of a projectile on the angle which it is fired, the height of the projectile above the landing area and its initial velocity.  The experimental range value of the projectile can then be compared to the calculated theoretical range value from the equations used to predict two-dimensional projectile motion.  The purpose of this lab is to verify the equations used to predict two-dimensional motion within the uncertainties of the experiment.

 

PROCEDURE

 

            The experiment was conducted by a group of students divided into smaller groups to complete the data collection.  The group members are William Jones, Mike Wilson, Mitch Moffet, Jennifer Alameda, Emily Stone and Annekathrin Friedrichs.  Each of the smaller groups was responsible for launching and measuring the projectile landing area, collecting the data and entering the data in an Excel spreadsheet according to the parameters outlined in the Physics 4A Laboratory Manual. 

            The first phase of the experiment consisted of setting up a Cenco Projectile Launcher, similar to the one shown in Figure 1.  The launcher was clamped to a board clamped to the table edge using C-clamps.  The launcher was clamped to the board using the launcher clamps.  The first series of launches was done at an angle of , set using the protractor mounted to the bottom of the projectile launcher.

 

Figure 1: Cenco Projectile Launcher

           

            The launcher spring firing position was set and checked for adequate medium initial velocity, enabling control over the distance and height the projectile was fired.  The initial velocity of the setting is unknown and must be calculated from the experiment results.  The initial height of the projectile was measured.  Several test shots were fired to locate the probable landing area of the metal ball.  Several pieces of white paper were taped to the floor in the landing area with carbon paper taped over the white paper.  The projectile was fired from the launcher a total of fifty (50) times striking the papered landing area.  The carbon paper was removed and the distance from the launcher the projectile landed for each launch was measured along the x-axis.  Figure 2 illustrates the landing area down range from the launcher.  

 

Figure 2: Projectile Landing Area

 

The projectile landed in a tight pattern with only a small deviation off the x-axis as shown in

Figure 3.  This deviation introduced a small error into the distance measurement, but not to a significant degree of error.

 

210 cm

Figure 3: Projectile Landing Pattern

Each hit on the paper was measured for its distance from the launcher using a wood 2-meter stick in conjunction with a meter stick and recorded in an Excel spreadsheet for later analysis.

            The next phase of the experiment consisted of five (5) projectile launches from several different angles, then ten (10) more launches from .  The launch angles were .  The initial height of the projectile was measured for each angle along with the change in the x-axis distance of each angle and recorded in the spreadsheet.  The distance the projectile was launched was recorded and measured in the manner described above and memorialized in the spreadsheet.

Three areas of uncertainty were introduced into the experiment from the human and mechanical setup.  Measurement of the distance was done within  0.5 cm accepting that the meter sticks were accurate and not warped.  As discussed previously, the projectile did not land on the x-axis in every instance.  The deviation in the “y” direction was not measured or considered.  The angle was set using the protractor mounted on the bottom of the launcher and was within  1 degree.   

 

ANALYSIS

 

A complete analysis of the data was done using an Excel spreadsheet including the graphs that were requested in the Physics 4A Laboratory Manual Lab 4 section.  Most of the spreadsheet is recreated in this report.  To view the spreadsheet analysis in its entirety click on the Excel hyperlink or graph.

 

PHASE I: DETERMINING INITIAL VELOCITY

 

            The fifty launches from a height of 90 cm (0.9 m) and  were measured and recorded in the Excel spreadsheet in cells A1 through A50.  This is the basis for the entire initial velocity analysis.  The projectile landed between 211.6 cm and 226 cm from the launcher with a delta R of 14.4 cm.  The average range distance “R” was 218.548 cm (2.18548 m).  The standard deviation calculated by the Excel program was 3.436255675 cm.

 

Spreadsheet Average, Standard Deviation, Low and High Values

 

            To see if the values of “R” are truly randomly distributed about the mean value, ten bins were set with the number of times the projectile landed in that bin.  The frequency vs. landing area was graphed using this data as shown in the graph Projectile Landing Range.  Note that thirty-one times the projectile landed between 215.8 cm and 221.4 cm.  The standard deviation covers an area between 215.1 cm to 221.9 cm.  The number of times the projectile landed in this area as shown by the graph is at least 33 plus times.

 

 

 

            To see if this graph is Gaussian in shape and if the standard deviation could be determined from the range shown above, a second line graph was created.  The formula for determining the standard deviation was taken from the Physics 4A Laboratory Manual.  

 

 

The maximum value of the curve N_max = 10.

Using the   formula , therefore: .  The points at 6.06 on the graph are at the landing area of 215.8  217.2 and just short of 221.4  222.8 cm.  One standard deviation is found from these points using the formula .  The calculated standard deviation was 3.4362… cm from the Excel spreadsheet.  The estimated value of 3.5 cm from the above graph is consistent with the calculated value.  The values do not match exactly because the exact range value where the line crosses the curve is not shown.  A smaller value less than 222.8 cm will lower the graphical standard deviation to a value closer or below the calculated standard deviation.  The standard deviation value calculated using the spreadsheet is used for . 

            The initial velocity, , are derived using the data collected for R_ave and h of 218.548 cm and 90 cm respectively. The initial velocity is:  

 

 

The theoretical range value for the projectile is calculated using the average range distance velocity along with the change in velocity calculated from the above figures.  The range formula used to calculate the theoretical range value is derived from the equations of motion in two dimensions given below.

 

 

motion in two dimensions equations

 

The horizontal component of the velocity is constant with a_x= 0.  The initial point is set at coordinates (0,0) making y_i and x_i equal to zero (0).  This makes x= R= v_ixt.

 

t= t_up + t_down or the time for the projectile to reach its maximum height then fall to its landing point.  The acceleration in the “y” direction is just that of the earth’s gravity “g” or 9.8 m/s².  As the projectile rises to its maximum height the velocity decreases to 0 m/s.  At its maximum height, its velocity is zero.  Therefore:   t_down is the time it takes the projectile to fall from its maximum height.

Where:  

 

 

 Therefore:  

 

 

Determining Range as a function of firing angle

 

Using the equation derived above, the theoretical range value for each angle the projectile was launched from was calculated using the Excel spreadsheet. 

 

 

Note:  The theoretical range value at zero degrees, or horizontal launch, is equal to the average distance used to calculate the initial velocity.

 

The experimental range value in cm measured for each launch angle is shown below.

 

 

 

The data from the theoretical range value and the experimental range value was compiled and put in the graph below for comparison of the two values.

 

 

This chart compares the range values from the experiment minimum, maximum, and average with the calculated range value at the initial velocity derived from the original average range values.  The theoretical range value is greater than the experimental values at .  The theoretical distance is lower than the experimental distance at .  There may be several causes for this discrepancy.  The first, and most probable, is the correction value measurement made to the x-axis as the launcher’s height increased with the angle.  It became more difficult to accurately measure the additional “x” distance increase using the plumb bob and ruler at greater angles.  The distance increase for  is out of line with the other increases for a rise in the angle of .  The x-axis distance increase was in the range of approximately 1.5 to 2 cm up to  then jumped by 5 cm for the  angle.

            The uncertainty in the velocity was calculated using the standard deviation of “R” to give a minimum and maximum theoretical range value of “R.”   This was found using the propagation of error formula.

 

 

 

            The maximum and minimum theoretical values of “R” were calculated using the above formula to calculate the change in “R” then added or subtracted to the theoretical value of “R.”

 

 

 

The following data table was created using the maximum and minimum values of “R.”

 

 

 

 

 

            This chart compares the minimum and maximum range value of the experiment with the theoretical minimum and maximum range values for the calculated initial velocity.  There were ten launches at  and five launches from each of the remaining angles.  For launches between , the experimental range values, min and max, fall within the theoretical minimum and maximum range values.  For launches between  the experimental minimum and maximum range exceeded the theoretical range.  In the cases of  the five launches fell within 10% of the theoretical range values.  In the case of  the projectile landed within 3 cm as the maximum and 2 cm as the minimum.  As stated earlier in this report, the protractor angle measurement was estimated to be .  This would certainly account for the projectile landing at a position of greater or less than the theoretical value.

An upward change in the projectiles velocity would seem to account for the experimental distance being greater than the theoretical distance.  The evidence does not support this theory.  The projectile launches from  preceded the final ten launches from horizontal.  As shown in the table below the range distance of where the projectile landed decreased from the original fifty launches.  The average initial velocity decreased from 5.0994533 m/s to 4.991 m/s.  The average range distance went from 218.5 cm to 213.9 cm, a decrease 4.6 cm.    The decrease can be attributed to several causes.  One is, that the number of launches is considerably less by a factor of 5.  Another is, the handling of the projectile launcher as the experiment progressed.  There were no intermediate checks of the launcher to ensure that its spring position remained in the exact initial position.  If the spring is starting to fatigue as the experiment progresses than the only explanation for the experimental range value increasing over the theoretical is an error in the launch angle greater than one degree or a measuring error from the increased angle adding an excess distance for a correction factor.

 

 

 

The chart shows that the maximum theoretical projectile distance is at  with a distance of

3.509 m.  At  the range value is very close to the  value.  This distance was calculated to be 3.497 m.  The experimental minimum and maximum range values are also very close to each other for the two angles.  Closer study of the experimental value data table shown above reveals that the two values are only close in their minimum and maximum values.  The minimum value for  was 326 cm.  The other four launches landed closer to 3.39 m.  At  the maximum range value was close to the higher values of the  launch at 338 cm, but the other four launches were closer to 330 cm.  In this experiment the projectile traveled its maximum distance at a launch angle of . 

 

 

The experiment did not include launches from several angles between the  to determine the exact experimental angle which the projectile will travel its maximum distance.  The experimental and theoretical values derived from the experiment clearly indicate that the maximum projectile distance will fall somewhere between .  Setting the height at the average height, between the two angles, of 0.984 m and using the formula for calculating the theoretical distance for each degree of angle from  the maximum range distance of the projectile was derived, as shown in the table and chart below.  The angle at which the projectile should travel its greatest distance is at .

 

 

 

 

 

 

 

            Overall the experimental range values were consistent with the theoretical range values using the calculated initial velocity from the average distance that the projectile traveled when launched from various angles at  increments, showing that the equations used to predict two-dimensional projectile motion are correct and do work within the uncertainties of the experiment.

 

Excel Spreadsheet