Revised 2/07. Projectile Motion

Transcription

1 LPC Phsics Reised /07 Prjectile Mtin Prjectile Mtin Purpse: T measure the dependence f the range f a prjectile n initial elcit height and firing angle. Als, t erif predictins made the b equatins gerning prjectile mtin within the eperimental and theretical uncertainties inherent in this eperiment. Equipment: Ballistic Gun Apparatus Prjectiles Shelf Bards, Clamps -meter Stick Carbn Paper, Scratch Paper, Masking Tape Graphical Analsis Sftware Ther: A prjectile fired frm a height h with initial elcit at an angle θ ο abe the hrizntal will trael a distance R gien b the fllwing equatin: 1 f 7

2 LPC Phsics Reised /07 Prjectile Mtin R (,, h) sin sinθ h θ = csθ + + g g g θ Eq. 1 Nte that if h = 0 in Eq. 1 then R sin g θ = Eq. and if θ ο = 0 in Eq. 1 then h R = Eq. 3 g s that the initial elcit,, can be determined frm the relatin: g R h = Eq. 4 if bth R and h are knwn. f 7

3 LPC Phsics Reised /07 Eperiment: Prjectile Mtin Part A: Determining the Initial Velcit 1. Clamp the prjectile gun securel n the table s that a prjectile ma be fired withut bstructin.. Select a latch setting n the gun that will allw its range t be near the maimum aailable distance. Set the angle near 30 degrees and fire the prjectile. If it traels t far r nt far enugh, then adjust the setting. Als, set the prjectile at 90 and be sure that the prjectile desn t strike the ceiling (therwise, u ll need t cnduct the eperiment utside, r use a shrter range setting). Chse ne range setting (shrt, medium, r lng) and use this setting fr the entire eperiment. D nt change this setting. 3. Set the eleatin angle t zer and measure the height f the prjectile (bttm t flr). 4. Cck and fire the prjectile a few times and bsere where it lands. Tape a piece f paper (centered n the aerage psitin) n the flr at this pint and cer it with a piece f carbn paper. 5. Fire the prjectile 5 times (i.e. it must strike the paper 5 times). If ne piece f paper desn t cer the range f impacts, tape additinal papers dwn. Stra prjectiles can be ignred. Nte that u will repeat this prcedure at the end f part B, s pa attentin t the lcatin f ur carbn paper. 6. Reme the paper and measure the distance that each prjectile lands frm the base f the gun. Recrd these distances n paper and, later, in a Graphical Analsis spreadsheet. Part B: Determining Range as a Functin f Firing Angle 7. Raise the gun t an angle f 15, 30, 40, 45, 60, 75 and 90 and fire the prjectile t see where it lands fr each angle. Place a sheet f paper and carbn paper in the target area, and fire at least three times frm each setting. Recrd the alues f R n paper, and later, in ur spreadsheet. Nte: Be careful t place ur gun s that it fires t ur right while facing it. 8. Nw set the angle f the launcher back t 0 and repeat step 5. Analsis: 1. Using Graphical Analsis, pen up a spreadsheet and enter the 5 alues f R that u measured in part A. 3 f 7

4 LPC Phsics Reised /07 Prjectile Mtin. Using the alues f R, determine R ag and σ R, where σ R is the standard deiatin in R. Of curse, u shuld actuall calculate these using GA s statistical analsis functin (t d this: Yu must create a plt f ur data. It des nt need t be srted r in an particular rder. In the X clumn, enter numbers 1-> 50(r hweer man data pints u end up with). In the Y clumn, enter ur R alues. GA will autmaticall create a graph f Y s. X. Chse Analze > Statistics, and a windw will appear n the graph giing u the aerage, ma, and min R alues, and the standard deiatin in R.). 3. Repeat steps 1- using the 5 alues f R that u measured fr 0 during part B. If ur alues fr the aerage and standard deiatin are cnsistent, cmbine all ur data int ne set f 50 alues fr use in analsis steps Nw u will see if the alues f R are trul randml distributed abut the mean alue. T d this, u will create a histgram f ur data. A histgram creates bins f data, and cunts hw man data pints fit int each bin (this number is called cunts r frequenc ). The graph then displas frequenc s. bin. T create a histgram f ur data, chse Insert > Additinal Graphs > Histgram frm the tp bar menu. Yu want t create 10 data bins, s u need t tell the prgram hw wide t make the bins. Yur bins are f distance traeled b the prjectile, s the bin width wuld be (d ma d min )/ In this srt f statistical eperiment, data ma be fudged b creatie cnstructin f binning. Yu knw u hae fund a gd bin size if changing the sizing up r dwn des nt change the shape f the histgram. T check ur binning, quickl create tw mre histgrams, ne with 11 bins, the ther with 9. Is either cure drasticall lpsided? If nt, gd! If s, u need t find a better bin size t wrk with...see ur instructr fr help. Once u hae determined a gd bin size fr ur data, keep that histgram and delete the rest. 6. Is ur graph Gaussian in shape? If s, u shuld be able t find the standard deiatin b nting that the equatin fr a Gaussian cure is: N ( sma ) σ = N e Eq. 5 ma Since this equatin describes the cure, that means that when ma = σ, then: 1 mae. 606 N = N = 0 N ma Eq. 6 Eample: 4 f 7

5 LPC Phsics Reised /07 Prjectile Mtin The graph belw is a Gaussian cure centered n the alue ma = 15. The maimum alue f this cure is N ma = 1. At ne standard deiatin frm the maimum, =.606*1 = 7.8. At this pint = 10.5, and 19.5 (apprimatel). Thus ne standard deiatin can be fund b: σ = = 9, r σ = Using this relatinship described abe, determine the alue f σ Ρ frm ur graph. Des ur alue f σ Ρ frm the graphical methd match the statistical methd? If nt, wh nt? 8. Determine the eperimental alue f and its uncertaint δ using the relatins: = g R ag h Eq. 7 δr δ = Eq. 8 R ag In this case, δr is the standard deiatin in R, (σ Ρ ). Unless ur graphs indicate a result t the cntrar (i.e. a nn-gaussian cure), use R and σ Ρ. Part B: 1. Derie Eq. 1, starting frm the equatins f mtin in tw dimensins (belw). Include the deriatin in ur lab reprt. 5 f 7

6 LPC Phsics Reised /07 Prjectile Mtin = = + = + a t + 1 a ( t ),, + a t, = = = + + a t + a + 1 a t ( ) Hint: Since the hrizntal cmpnent f the elcit is cnstant, (t) = R = t where t = t up + t dwn. Net, cnsider that when the prjectile reaches it maimum height, then = 0. Als, t dwn is simpl the time it takes fr the prjectile t free fall frm its maimum height.. Using the alue f u determined in Part A, calculate the theretical alue f R as gien b Eq. 1, fr each angle θ and crrespnding height h. Use ur spreadsheet t perfrm this calculatin. T test ur equatin, perfrm the calculatin fr θ = 0, and θ = 90. The results shuld be R ag (frm Part A) and 0 respectiel. Nte: A cmmn mistake in this calculatin is t use radians rather than degrees. 3. T displa ur results, plt a bar graph f R ma, R min, R ag, and R ther s. firing angle. Yu ma need t rearrange ur spreadsheet data clumns t make this wrk. 4. There is ne last lse thread befre u tie all the data up in a neat package. The uncertaint in elcit (calculated b the standard deiatin in R) causes significant uncertaint in the theretical alue. This can be fund b the prpagatin f errr technique: dr δ Rther = δ d In practice this can be a daunting task, especiall gien the cmpleit f equatin. Hweer, it can be simplified b nting that the first tw terms in equatin (1) depend n and are thus mre effected b uncertaint than the term inside the square rt (which depends nl n the first pwer f ). This is equialent t cnsidering the range f a prjectile fired frm a height h = 0. Thus, R = sin(θ ) g. The deriatie with respect t is simpl dr/d = sin(θ)/g. Calculate δr ther fr each alue f θ. Create tw mre clumns n ur spreadsheet. One will be the maimum theretical range alue (R ther + δr ther ), and the ther will be the minimum theretical range alue (R ther - δr ther ). Create a bar graph as befre, displaing the ma and min alues f R ther alng with the maimum and minimum alues f R ep. 6 f 7

7 LPC Phsics Reised /07 Prjectile Mtin 5. Interpret the results f the graph in Step 3 and discuss an anmalies that ma be present in the data. 6. At what launch angle did the prjectile trael its maimum distance? Wh was this angle nt equal t 45? At what angle d u epect the maimum distance (hint: this might require sme calculus)? Include ur answers in ur lab reprt. Results: At the end f ur reprt, write at least ne paragraph describing the fllwing: 1. what u epected t learn abut the lab (i.e. what was the reasn fr cnducting the eperiment?). ur results, hw the cmpared with ur epectatins, and what u learned frm them ptinal but usuall a gd idea 3. Think f at least ne ther eperiment might u perfrm t erif these results 4. Think f at least ne new questin r prblem that culd be answered with the phsics u hae learned in this labratr, r be etraplated frm the ideas in this labratr. 7 f 7