Introduction to Kinematics (Constant Velocity and Acceleration)

Transcription

1 Inroducion o Kinemaics (Consan Velociy and Acceleraion) Inroducion To race moion of an objec, we have o know how i moves wih respec o ime. Namely, i is expeced o record he change of moion in erms of elapsed ime. The moion of an objec can be described by he displacemen, velociy, and acceleraion. Each of hem has a dimension [L], [L][T -1 ], and [L][T - ], respecively; namely, each uni has o be m, m/s, and m/s. Displacemen indicaes he direcion and disance from he iniial and he final poins. Velociy is he change of a displacemen in a cerain ime inerval. Acceleraion is a change of velociy in cerain ime. Thus, a consan acceleraion creaes a consan increase of he velociy and a consan velociy creaes a consan change of he displacemen. This can be found in daily life experiences such as raveling wih a car. On he oher hand, he mos common consan acceleraion in naure is known as graviaional acceleraion. Objecs on he Earh fall oward he cener of he Earh wih an acceleraion, 9.81 m/s. This depics ha a falling objec obains velociy, 9.81 m/s, each second. The procedure of he experimen o obain graviaional acceleraion is following: Under a consan acceleraion, he average velociy, v, is (final velociy + iniial velociy) divided by. Tha is: v f + v i v = Making he iniial velociy zero, we obain he average velociy as only final velociy divided by. Anoher expression of he average velociy is disance (or heigh) divided by ime: h v = From hese equaions, he average velociy is eliminaed and we have: v f h = Thus, he final velociy can be solved as h v f = Therefore, he acceleraion reaching he final velociy is he final velociy divided by he ime, v f ; namely, h a = The acceleraion, a, from he free-fall experimen is supposed o be he graviaional acceleraion, g, 9.81 m/s. Objecives: To learn qualiaive descripion of moion (consan velociy and acceleraion) 1. A consan To verify acceleraion he graviaional acceleraion as 9.81 m/s (consan falling acceleraion)

2 Procedure: Se up he following experimenaion as shown: Sar up DaaSudio and selec he moion sensor. [Click a channel, A or B, in he inerface of he picure only ONCE alhough here are wo connecions.] Open wo graphs of velociy vs. ime and acceleraion vs. ime. 3 Mimic a siuaion when you ge on a highway o reach 70 mi/h wih he miniaure se up. Aach a sring o he car if needed and pull i by imagining he siuaion. (Le s say, accelerae uniformly from res o 0.5 m/s in his case.) Then, obain wo graphs wih he DaaSudio. 4 When you wan o delee he previous daa, go o Experimen and selec eiher Delee Las Daa Run or Delee ALL Daa Runs as shown. Make sure: The graphs likely give you he following resul. Try o obain he expeced resuls. Also, le everyone ry his experimen. Concepual noe: The above picure illusraes an ideal case and you would mos likely obain he following graph: Does i make sense? Explain wha happened o he velociy afer reaching he peak.

3 Prin his ou! Check wih your insrucor o see if you obained he proper graphs wih DaaSudio. One bes graph is enough o share wih oher parners. [Do NOT modify any daa. Use every daa poin if i physically makes sense.] Wha is acceleraion? How do he velociy and acceleraion change wih respec o ime? Wha is he meaning of consan acceleraion according o he graph?. A consan velociy Procedure: Use he same se up as previously. 3 Mimic a siuaion when you have o drive a 70 mi/h consanly (no acceleraion, no o be caugh by police) wih he miniaure se up. Aach a sring o he car if needed and pull i by imagining he siuaion. Then, obain wo graphs wih he DaaSudio. 4 When you wan o delee he previous daa, go o Experimen and selec eiher Delee Las Daa Run or Delee ALL Daa Runs as shown. Make sure: The graphs likely give you he following resul. Try o obain he expeced resuls. Also, le everyone ry his experimen. Concepual noe: The above picure illusraes an ideal case and you would mos likely obain he following graph: Does i make sense? Explain wha happened o he velociy before and afer reaching he plaeau. v An example of acual daa

4 Prin his ou! Check wih your insrucor o see if you obained he proper graphs wih DaaSudio. One bes graph is enough o share wih oher parners. [Do NOT modify any daa. Use every daa poin if i physically makes sense.] ❶Wha is he relaionship beween velociy and acceleraion? ❷How do you describe hese wo graphs? ❸How do you insruc your friends how o make a consan speed wih a car? 3. Graviaional acceleraion g: Procedure: Se up he pieces of equipmen as shown in he figure: The phoo gae works as follows: Sar up DaaSudio. Click he digial channels 1 o selec he phoo gae. Do he same for he digial channel. Click he able for he display. Then, he daa source has o be Time Beween Any Gaes as shown below: Click he digial channels o have wo phoo gaes. Click here and choose Time b/w any gaes. 3 Measure he fall disance, h, wih a meer sick, which is beween wo phoo gaes (beween infrared beams).

5 4 The fall ime is obained from he phoo gae. The iniial place o drop mus be as close o he phoo gae as possible because he iniial velociy is assumed as zero. This is very imporan o obain an accurae resul. The ips are following: Make he objec inerrup he infrared and he ligh will be urned on. Jus lif he objec righ above he beam. Afer making sure he ligh urned off, ell your parners o click sar. 5 The elapsed ime is wha you record in he daa able as shown: 6 According o he inroducion of his lab, we can assume ha he acceleraion is h consan. Using he average velociies, we have he final velociy as v =. Therefore, he acceleraion reaching he final velociy is h f a =. The acceleraion, a, has o be he graviaional acceleraion, g, 9.81 m/s. Do no ge confused wih a and g. The a is he acceleraion measured by he experimen, which is supposed o be close o he consan falling acceleraion, g, graviaional acceleraion. Hins: If you obained he acceleraion greaer han 9.81 m/s, you would likely make an iniial velociy by releasing he objec higher han i should have been. If you obained i less han he acceped value, he objec would block he phoo gae before dropping Fall disance, Fall ime, The final (Use he The acceleraion % Error h (by phoo gae) velociy a he h a = Column (m) disance value in 100 % Take Elapsed 9.81 (Columns 1 ) column.) Times. * = v i v f ; hus, v f = v since v i is zero. * v ( + )/ Quesion: How does he error depend on he fall disance? (The more disance, he more % error, or opposie? Discuss he reasons.) How do you describe your experimen in erms of he daa disribuion? (See secion 6 in he firs lab.)

6 Plo a graph using columns and 1. (Noe ha falling disance, h, is in y -axis and ime is in x -axis.) Quesions: According o column 3 above, is i a uniformly acceleraed, or a non-uniformly acceleraed moion? Are he resuls of he acceleraion close o he acceped values by referring o he column 5? Wha are he possible causes of errors? From he above graph, how does he falling disance vary wih ime? Is i linearly proporional?

7 The heory of he following analysis: The acceleraion formula is given as a =. Solving for h, we have h = a h. Compare i wih a line equaion, y = bx. The values,, and h correspond o x and y axes, respecively. Therefore, ploing a graph using columns 4 and 1 gives you he slope, which is he acceleraion, a. You obain i wih he linear fi line program in DaaSudio. The slope is supposed o be he graviaional acceleraion, g = 9.81 m/s. [Noe: The daa shown below are only for he explanaion purpose. Please do no compare your resuls wih hese.] Resar DaaSudio o have opening screen. Click Ener Daa. Type he daa in he daa columns as indicaed. For column 4 daa For column 1 daa 3 Click Fi in he panel, hen selec Linear Fi. 4 The linear fi line and is informaion will be displayed as shown. [The daa used in he figure are ficiious!] Quesion: From he slope of he linear fi line, how well does your resul agree wih he expeced value (9.81 m/s ) also in erms of he uncerainy? [Wha is (are) he advanage(s) of his averaging mehod?]

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