Kinematics in 2-Dimensions. Projectile Motion

Transcription

1 Knematcs n -Dmensons Projectle Moton A medeval trebuchet b Kolderer, c Readng Assgnment: Chapter 4, Sectons -6 Introducton: In medeval das, people had a ver practcal knowledge of projectle moton. The ma not have understood the exact trajector that a projectle would take, but b practce the could place a projectle on a target consstentl from a dstance of well over 00 ards. Durng a long sege of a castle, t was not uncommon to hurl bodes of anmals (and es, captves) back nto the beseged castle s water suppl (an earl form of bologcal warfare). Smlarl, a modern da hunter does not need to know the actual path that a bullet takes to a target n order to ht the target. A sharpshooter, however, does know the path and can make adjustments n the amng n order to ht a target at man dfferent ranges. In ths lab, ou wll become a sharpshooter of sorts. You wll use the equatons of moton to predct the path of a projectle and ht a target. Neglectng frctonal forces, such as ar resstance, an object projected from a launcher undergoes a moton that s the smple vector combnaton of unform veloct n the horzontal drecton and unform acceleraton n the vertcal drecton. For a projectle launched wth a speed, v, at an angle θ wth respect to the postve x axs, t can be shown that the trajector caused b such a combnaton predcts a parabolc shape. The followng knematc equatons descrbe ths moton: 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

2 Horzontal Moton: Vertcal Moton: x ( t ) = x + v x t ( t) = + v t a t Eqn. 1 1 Eqn. + v ( t) = x v x v ( t ) = v + a t Eqn. 3 v ( t) ( ( t ) = v + a ) Eqn. 4 where v and v x are the ntal vertcal and horzontal components of the veloct respectvel. Notce that Equatons 1 and have a common varable, t. Equaton 1 predcts the x coordnate n terms of the parameter t, Equaton predcts the coordnate n terms of the parameter, t. B combnng these two equatons, the dependenc upon the parameter, t, can be elmnated. Smpl solvng Equaton 1 for t and substtutng t nto Equaton results n the followng: ( ) 1 x = + + v v x a x v x x Where: x = x( t) x whch smplfes to: v a ( x) = + x x v + x ( v ) x ( ) Furthermore, the components of the veloct can be wrtten n terms of the orgnal launch veloct as: v = v cosθ v = v snθ x Eqn. 5 These components, when combned wth Equaton 5 eld an equaton for (x) determned completel b v and Θ (the ntal launch speed and angle): a ( x) = + tanθ x + x ( v cosθ ) Eqn. 6 Notce that Equatons 5 and 6 descrbe the poston of the object but the do not sa when (at what tme) the object has an partcular poston. Also, notce that the relatonshp between and t n Equaton s quadratc (parabolc) n t because the values for a, v, and are constant. Smlarl, n Equatons 5 and 6, vertcal poston,, as a functon of horzontal poston, x, s quadratc (parabolc) n x because the values for a, v, and v x (t) are also constant. The equaton for (x) represents the trajector of the projectle. If a, x(t), x, θ, and are known, then t should be possble to determne the speed at whch the projectle was launched. Note: The famous Range Equaton for projectle moton s a specal case of the dervaton descrbed above. It can onl be used when a projectle starts and lands at exactl the same vertcal heght. It also defnes a coordnate axs for the trajector such that x = 0 and = 0. ( ) 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

3 As an exercse, plot Equaton 6 as vs. x for a varet of realstc values for θ, v, and. What determnes the shape of the curve, the x poston of the maxmum, and the heght of the curve ( poston of the maxmum)? Expermentng wth the mathematcs of a trajector can eld tremendous nsght nto projectle moton. 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

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5 Knematcs n -Dmensons Projectle Moton Goals: Resolve veloct vectors nto components. Determne the muzzle veloct of a projectle launcher. Predct the range of a projectle. Use Excel to analze the moton of a projectle. Equpment Lst: Projectle Launcher Steel projectle Meter Stck or Tape Measure Table clamp Carbon paper Excel Paper Target Scotch Tape Lab Actvt 1: Determnng Launch Veloct (Tabletop to tabletop launch) 1. Set up a projectle launcher at an arbtrar angle, other than 0 or 90 o. The launcher should be adjusted so that t projects the projectle onto the tabletop. The angle that ou set, θ, s the angle that ou wll use throughout actvtes 1 and. DO NOT pont our launcher n the drecton of the computer montors!. Carefull measure the heght from the tabletop to the launchng poston of the projectle. The manufacturer has placed a mark on the sde of the launcher for the purpose of ths measurement. Ths s the ntal vertcal poston,. 3. The launcher has three ranges: each range s determned b a clck n the sprng launcher and s also marked on the sde of the launcher. Be sure to use the second clck (medum range settng). 4. Fre the launcher and have a lab partner note the approxmate poston that the projectle strkes the table. Tape a pece of paper to the tabletop and place a sheet of carbon paper (carbon sde down) on top of the taped paper. It s not necessar to tape the carbon paper to the table. 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

6 5. Fre for effect! Fre the launcher several tmes to obtan an average landng poston. Estmate the center poston of our pattern and measure the horzontal dstance from ths pont to a pont drectl below the launchng poston of the projectle. Ths s the horzontal range, x. 6. Use the values for a, x(t), x, θ, (x) and to calculate the speed, v, at whch the projectle left the launcher. (Assume that x = 0, and a = -g = -9.8 m/s.) 7. Record the values of θ and the calculated launch speed, v, for use n the next actvt. Lab Actvt : Determnng Projectle Range (Tabletop to floor launch) In the prevous actvt, ou determned the speed and drecton, hence the veloct, of a projectle beng fred from our launcher. You wll now use ths nformaton to predct the landng pont of a projectle that s launched from the tabletop to the floor. 1. Turn our launcher so that t faces awa from the table and towards an open space on the floor. Measure the vertcal dstance from the launchng pont of the projectle (on the sde of the launcher) to the floor. Ths wll be our new.. Use the value of v,, and θ to determne the horzontal dstance at whch a target must be placed n order to be ht wth our projectle. 3. Contact the nstructor when ou are read to fre our projectle. IMPORTANT: Do not take an shots before the TA s contacted or ou wll have our launcher angle changed and ou wll need to begn agan. 4. When the TA s read, fre for effect. Your score wll be the total ponts scored n fve shots. You wll be allowed one adjustment to the horzontal drecton of the launcher. 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

7 Lab Actvt 3: Range vs. Angle Theor predcts that, when a projectle starts and lands from the same vertcal heght, the maxmum horzontal range should occur at 45 o. In addton, there should be two dstnct angles (complementar angles) of launch that would send a projectle to a partcular range less than the maxmum range. 1. Fre our projectle launcher (so that the projectle agan lands on the tabletop) at dfferent angles from 0 up to but not ncludng 90 at ether 5 or 10 ncrements (dependng on the amount of tme avalable but nclude a data pont for 45 ). Poston the launcher so that ts plumb bob hangs off of the edge of the table. Ths wll enable ou to make measurements at large angles. Record data for range and launch angle n a table, such as: Launch Angle (degrees) Range (meters). Usng Excel, make a graph of Range vs. Angle. 3. Based upon our graph, does the maxmum range occur at 45? If not, where does t occur? 4. From our graph, generate an estmated lst of at least 5 pars of angle measures that eld the same range values? Is each par a set of complementar angles? Explan. 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons

8 Name Secton # Name Name 008 Penn State Unverst Phscs 11: Lab Knematcs n -Dmensons