2d Motion: Constant Acceleration


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1 d Moion: Consan Acceleaion Kinemaic Equaions o Moion (eco Fom Acceleaion eco (consan eloci eco (uncion o Posiion eco (uncion o The eloci eco and posiion eco ae a uncion o he ime. eloci eco a ime. Posiion eco a ime. a a ˆ a ˆ ( a ( a ˆ ˆ The componens o he posiion eco a, and, ae consans. R. Field 9/6/ PHY 53 Pae Uniesi o Floida ˆ The componens o he acceleaion eco, a and a, ae consans. The componens o he eloci eco a, and, ae consans. Wanin! These equaions ae onl alid i he acceleaion is consan. ˆ
2 a d Moion: Consan Acceleaion Kinemaic Equaions o Moion (Componen Fom consan ( a ( a a The componens o he acceleaion eco, a and a, ae consans. The componens o he eloci eco a, and, ae consans. The componens o he posiion eco a, and, ae consans. consan ( a ( a Wanin! These equaions ae onl alid i he acceleaion is consan. Ancilla Equaions ( o a( ( a ( ( ( o alid a an ime R. Field 9/6/ Uniesi o Floida PHY 53 Pae
3 Acceleaion Due o Gai Epeimenal Resul Nea he suace o he Eah all objecs all owad he cene o he Eah wih he same consan acceleaion, 9.8 m/s, (in a acuum independen o mass, sie, shape, ec. ais h ais Equaions o Moion a ( 9.8m The acceleaion due o ai is almos consan and equal o 9.8 m/s poided h << R E! R E ( Eah ( R. Field 9/6/ Uniesi o Floida PHY 53 Pae 3
4 Pojecile Moion Nea he Suace o he Eah In his case, a and a , cosθ,,, h. a ( cosθ ( ( cosθ a (consan ( ( h ( θ sin ais θ h ais Maimum Heih H The ime, ma, ha he pojecie eaches is maimum heih occus when ( ma. Hence, H ma ( ma h ( Fo a ied he laes H occus when θ 9 o! R. Field 9/6/ Uniesi o Floida PHY 53 Pae 4
5 R. Field 9/6/ Uniesi o Floida Pojecile Moion Nea he Suace o he Eah (h In his case, a and a , cosθ,,,. cosθ ( ( ( cosθ Maimum Heih H ( ( ( The ime, ma, ha he pojecie eaches is maimum heih occus when ( ma. Hence, ( ma H ( ma Rane R (maimum hoional disance aeled The ime,, ha i akes he pojecie each he ound occus when (. Hence, ( ( sin θ R ( ( cosθ cosθ sin θ ais ais PHY 53 Pae 5 θ
6 R. Field 9/6/ Uniesi o Floida Pojecile Moion Nea he Suace o he Eah (h In his case, a and a , cosθ,,,. cosθ ( ( ( cosθ Maimum Heih H ( ( ( The ime, ma, ha he pojecie eaches is maimum heih occus when ( ma. Hence, ( ma H ( ma Rane R (maimum hoional Fo a ied he laes disance R aeled occus when θ 45 The ime,, ha i akes he pojecie eachhe ound o! occus when (. Hence, ( ( sin R θ ( ( cosθ cosθ sin θ ais ais PHY 53 Pae 6 θ
7 Eample Poblem: Pojecile Moion A suspension bide is 6. m aboe he leel base o a oe. A sone is hown o dopped o he bide. Inoe ai esisance. A he locaion o he bide has been measued o be 9.83 m/s. I ou dop he sone how lon does i ake o i o all o he base o he oe? In his case, a and a ,,,, h. Hence, ( ( ( ( h The ime,, ha he i akes he sone o each he ound occus when (. Hence, ( h h (6m s (9.83m I ou how he sone saih down wih a speed o. m/s, how lon beoe i his he ound? Hae o use he In his case, a and a ,, ,, h. Hence, Quadaic Fomula! ( ( h ± ( o h m ± ( m (9.83m (6m. s 9.83m ais h ais R. Field 9/6/ Uniesi o Floida PHY 53 Pae 7
8 R. Field 9/6/ Uniesi o Floida PHY 53 Pae 8 Reeence Fames Conside wo ames o eeence he Oame (label eens accodin o,,, and he Oame (label eens accodin o,,, moin a a consan eloci, wih espec o each ohe a le he oiins coincide a. In he Galilean ansomaions he O and O ames ae elaed as ollows: Galilean eloci Tansomaion: Een O: (,,, O: (.,, O O ( Time is absolue! Classical eloci addiion omula!
9 Posulaes o Classical Phsics Fis Posulae o Classical Phsics ( Relaii Pinciple : The basic laws o phsics ae idenical in all ssems o eeence (ames which moe wih uniom (unacceleaed eloci wih espec o one anohe. The laws o phsics ae inaian unde a chane o ineial ame. The laws o phsics hae he same om in all ineial ames. I is impossible o deec uniom moion. O O Een O: (,,, O: (.,, Second Posulae o Classical Phsics (Galilean Tansomaion: The O and O ame ae elaed b he Galilean Tansomaion. Classical eloci addiion omula! R. Field 9/6/ Uniesi o Floida PHY 53 Pae 9
10 eloci Addiion Theoem eloci Addiion Theoem (eco om: AB BC AC BC AC A elaie o B B elaie o C A elaie o C AB Eample Poblem: Jack wans o ow diecl acoss a ie om he eas shoe o a poin on he wes shoe. The widh o he ie is 5 m and he cuen lows om noh o souh a.6 m/s. The ip akes Jack 4. min. In wha diecion did he head his owboa o ollow a couse due wes acoss he ie? A wha speed wih espec o he sill wae is Jack able o ow? 5m RW WS RS RS.99m RS (4. min(6s / min W N S ais ais E R. Field 9/6/ Uniesi o Floida anθ o θ 3.59 RW WS RS.6m.99m Noh o Wes RS ˆ WS.649 ˆ RW (.99m.6m ( ( (.6m PHY 53 Pae RW RS WS
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