Lecture : Single-particle Mtin Befre we start, let s l at Newtn s 3 rd Law Iagine a situatin where frces are nt transitted instantly between tw bdies, but rather prpagate at se velcity c This is true fr any real frce in nature Assue tw bdies are at rest, exerting attractive frces n each ther F 1 F 1 1 d Nw assue bdy 1 ves instananeusly t a new psitin We nw this can t really happen withut an infinite frce being applied, but tae it as an apprxiatin
Befre a tie t = d/c has passed, the frces nw l lie this: Bdy has yet t ntice F 1 that bdy 1 has ved. Since bdy 1 is ving thrugh the static frce field F 1 generated by, it always feels 1 a frce tward. The tw frces are n lnger equal r ppsite Newtn s Third Law really nly wrs in static situatins A hint that there s re t physics than just Newtn! Since c is large cpared t the velcities f the bjects we ll be cnsidering, we usually tae Newtn s Third Law as a gd apprxiatin
Mtin f a Single Particle Nw we apply Newtn s Laws t the tin f a single particle Prcedure fr slving prbles is: 1. Mae a diagra indicating all frces acting n the bject. Set up a cnvenient crdinate syste e.g., if the tin is alng a line, chse ne axis t lie alng that line 3. Use Newtn s Laws t deterine each cpnent f the acceleratin 4. Slve the differential equatin t deterine velcity and psitin During this step, ne taes int accunt the initial cnditins, such as initial psitin r velcity
Exaple 1: Cnstant frces If the frce acting n a particle is cnstant, the equatin f tin is: r = F In this case, we can always chse ne f the axes (say x 1 ) t lie in the directin f the acceleratin: dv1 x = F = 1 dt This differential equatin is easy t slve by integratin: Fdt = dv1 F v1 ( t) = t + v = v + at 1, 1,
One re integratin gives us the psitin as a functin f tie: dx1 = v + at 1, dt v 1, and x 1, are F cnstants dx = t v 1 + dt 1, deterined by the initial cnditins f 1 the prble x1 ( t) = x + v t + at 1, 1,
Prjectile Mtin One exaple f an (apprxiatly) cnstant frce is gravity near the Earth s surface F = g F a = = 9.8/s gˆ In this case, nt nly is the frce cnstant, but the acceleratin is the sae fr all bjects
Any (unpwered) bject flying thrugh the air near the earth s surface is called a prjectile If we tae the x axis t be hrizntal and the y axis vertical, we have: 1 x( t) = x + v t + a t = x + v t x, x 0 x, 1 1 y( t) = y + v t + a t = y + v t gt y, y y, Can als find y(x): x x 1 x x y( x) = y + v g y, v v x, x, s the trajectry is described by a parabla
Range f a Prjectile We can deterine the range f a prjectile fr a given initial velcity and launch angle by seeing when it returns t its initial height: R 1 R y( R) = y + v sinθ g y v csθ = v csθ sinθ 1 R = g v θ v R = = g csθ cs sinθ csθ v sin g θ
Exaple: Hitting a He Run Prjectile tin is very iprtant t the gae f baseball Equatins deterine whether a batted ball will be a he run, r just a flyut Assue a batter gives the ball an initial velcity f 100ph, and initial angle f 45. Hw far des the ball fly? ( 144ft/s) v sin θ R = = = 648ft g 3ft/s Oddly, thugh, the typical distance required fr a he run is ~400ft Is the gae f baseball just t easy? Or are we neglecting an iprtant effect?
Resistive Frces Fr a re realistic descriptin f prjectile tin, we need t include the effects f air resistance This frce clearly increases as the bject ves faster thrugh the air Actual dependence n velcity can be cplicated e.g., frce increases drastically as bject nears the speed f sund We ll tae a siple del fr this, and assue that the frce increases linearly with the velcity Newtn hiself assued the frce increased with the velcity squared, which is actually a re easily-justified assuptin
The equatins f tin then bece: dvx x = x = v x dt dvy y = g y = g v dt which again can be slved by integratin: dvx dvy = dt = dt v x g + vydt ln vx = t + C lng + v = t + C 1 y t g g vx = vx, e vy = + v y, + e y t
We see that the behavir as t is as we wuld expect: g vx 0; vy cnst = Terinal velcity One re integratin gives us the psitin as a functin f tie: dx t dy g g = vx, e = + v y, + e dt dt t g t g x = vx, e + C y = t v y, + e + C t g g t x = x + vx, 1 e y = y t + v y, + 1 e t
Let s again find the range f the prjectile Rather than writing y in ters f x, we can slve fr the tie f flight, and plug that value in t the equatin fr x g g t F y ( tf ) = y = y tf + v y, + 1 e 1 g tf = v + e g t F y, 1 This is a transcendental equatin, eaning there is n exact slutin Hwever, as physicists, we re nt allwed t give up! We need t find the best apprxiate slutin that we can Of curse, it s easy t d n a cputer, but if we re stuc n a deserted island (r ding P31 hewr).