I. Understand get a conceptual grasp of the problem

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departent o Physics Physics 81T Fall Ter 4 Class Proble 1: Solution Proble 1 A car is driving at a constant but unknown velocity,, on a straightaway A otorcycle is a distance d behind the car Initially, they are both traveling at the sae velocity The otorcycle starts to pass the car by speeding up at a constant acceleration a When the otorcyclist is side by side with the car, the otorcycle stops accelerating and is traveling at twice the velocity o the car How long does the otorcycle accelerate? What was the initial velocity o the car and otorcycle? How ar did the otorcycle travel while accelerating? Express all your answers in ters o the given quantities in the proble I Understand get a conceptual grasp o the proble We assue you ve recognized the proble is in the doain o kineatics the quantitative description o otion What is the proble asking? What are the given conditions and assuptions? What is to be ound and how is this deterined or constrained by the given conditions? In particular: how any objects are there, is the otion in 1,, or 3 diensions, is the otion relative or is there soe logical absolute reerence rae? Qualitatively describe the otion (in each coordinate and o each body i there are several) Model: Look or the three ost coon odel otions either the velocity or the acceleration is constant or the otion is unior circular Oten these apply only to part o the tie (or to only one body) ie ight the acceleration be constant but dierent beore and ater the rocket engine stops or between the two racing cars? Is the otion an exaple o unior circular otion Advice: Write your own representation o the proble s stated data: draw a otion diagra (strobe picture), a graph o position or velocity or acceleration vs tie, or a diagra Make a table o these quantities vs tie i it s nuerical What are the initial conditions and how do you represent conditions atheatically (eg until car A passes car B) A great any probles will involve special otion, perhaps in one or another coordinate: constant velocity, constant acceleration, unior circular otion, relative otion learn to recognize these otions Get the proble into your brain! Question: Describe the strategy you have chosen or solving this proble You ay want to consider the ollowing issues What does a sketch o the 1

2 proble look like? What type o coordinate syste will you choose? What inoration can you deduce ro a plot o distance vs tie or both the car and the otorcycle? What conditions ust be satisied when the person just catches up to the streetcar? Answer: Note: Your answer should include a sketch and coordinate syste or the syste where you clearly indicated your choice o origin, positive directions and reerence rae; a single graph showing qualitatively the position o the otorcycle and the driver as a unction o tie Modeling the proble: This is a two body proble with each body undergoing one diensional otion First I sketch the proble and introduce a Cartesian coordinate My irst question is where should I choose y origin I will choose t = to depict the oent the otorcycle is at the origin and the car is at a positive distance d At an arbitrary tie, I introduce the position unctions x 1 () t or the otorcycle and x () t or the car I denote the tie the car overtakes the otorcycle as t Figure 1: Coordinate Syste I a speciically interested in the tie period that the otorcycle is accelerating at a constant rate while the car is oving with a unior velocity This eans that I will need two sets o kineatic equations or position and velocity or the otorcycle and car noting that the acceleration o the car is zero, a = I can relate these sets o c equations by two extra conditions that are stated in the proble At the oent the otorcycle overtakes the car, the positions are equal and the otorcycle velocity is twice the car (igure )

3 Figure : plot o position vs tie or car and otorcycle The key is to translate all this inoration into speciic equations II Devise a Plan - set up a procedure to obtain the desired solution General - Have you seen a proble like this ie does the proble it in a schea you already know? Is a part o the proble a known schea; could you sipliy this proble so that it is? Can you ind any useul results ro the given initial conditions and other data even i it is not the solution? Can you iagine a route to the solution i only you know soe apparently not given inoration? Count the unknowns and check that you have that any independent equations In particular: choose the best type o coordinate syste to sipliy the proble, pick the orientation and location o the origin o the coordinate syste in accord with the initial conditions Warning: alost always it is best to pick positive to the right or up and represent downwards acceleration, or exaple, as g Given that the proble involves soe particular type o otion (constant acceleration, circular otion) think over all the equations that involve this concept Question: Devise a plan or solving or: how long does the otorcycle accelerate?; what was the initial velocity o the car and otorcycle?; how ar did the otorcycle travel while accelerating? Answer: I can write down the kineatic equations or the otorcycle, with the initial position corresponding to y choice o origin, x 1, = ; the initial velocity, the unknown v 1, = ; and the acceleration o the otorcycle a 1 = a ; then 3

4 x 1 ()= t v t + 1 at v () t = v + a t x,1 The initial position o the car is x, = d ; the initial velocity o the car is the sae as the otorcycle v, = ; and the acceleration o the car is a = ; so the kineatic equations or position and velocity o the car are Note that the velocity o the car is constant, x (t) = d + v t v x, () t = At the overlap tie, t, I can now state the two extra conditions atheatically The irst is that the position o the car and the otorcycle are equal, x 1 (t ) = x (t ) This becoes, using our kineatic equations or position, vt + 1 a t = d + v t I notice an iediately sipliication, so I rewrite this condition as 1 at = d My second condition at the overtake tie is that the velocity o the otorcycle is twice the velocity o the car, v x,1 ( t ) = v This equation becoes using the equation or the velocity o the otorcycle + a t = This equation sipliies to at = 4

5 Suary: I now collect y two equations, 1 at = d at = I now see that I have two equations and two unknowns, the tie o overlap, t, and the unknown initial velocity, v Although the distance and acceleration are not given speciic values they are speciied as given constants hence I will treat the representing sybols as known quantities Two independent equations with two unknowns are solvable So I just need to decide which quantity I will solve or irst Since the irst question asks how long does the otorcycle accelerate; I will solve or the tie o overlap, t Then I will solve or the unknown velocity, and inally or the distance the otorcycle travels using ( ) = v t + 1 x 1 t a t = t + d = x t ( ) III Carry our your plan solve the proble! This generally involves atheatical anipulations Try to keep the as siple as possible by not substituting in lengthy algebraic expressions until the end is in sight, ake your work as neat as you can to ease checking and reduce careless istakes Keep a clear idea o where you are going and have been (label the equations and what you have now ound), i possible, check each step as you proceed Solution: Fro the irst equation, tie o overlap, t, is t = d / a I will use this in the second equation to ind the unknown velocity = a t = a d / a = da The distance traveled by the otorcycle is then x ( t ) / + 1 = v t + d = ( da )( d a ) + d = d d = 3d IV Look Back check your solution and ethod o solution 5

6 Can you see that the answer is correct now that you have it oten siply by retrospective inspection? Can you solve it a dierent way? Is the proble equivalent to one you ve solved beore i the variables have soe speciic values? In particular: Check diensions i analytic, units i nuerical Check special cases (ie i a = does the solution sipliy?), check that a general expressions reproduce the given initial conditions Does it depend sensibly on the various quantities (eg is the tie greater i the initial velocity is less?)? Is the scaling what you d expect (tie decreases with the square root o the acceleration, distance at soe later tie proportional to initial velocity)? Is the answer physically reasonable (especially i nubers are given or reasonable ones substituted) Review the schea o the proble what is the odel, the physical approxiations, the concepts needed, and any tricky ath anipulation Question: Choose what you think are reasonable values or the distance d, and the constant acceleration a What values do you then calculate or how long the otorcycle accelerates?; what is the initial velocity o the car and otorcycle; and how ar does the otorcycle traveled while accelerating? Do your values ake sense to you? Answer: Units: My equation or the tie it took to overtake the car is t = d/ a The units (diensions) o tie are seconds, [di t ] = / s - = s, and velocity, = da, -1 has unit (diensions) [di v ] = ()( s - ) = s Both quantities have the correct units -1 Values: Suppose the car is oving at = s, and is a distance d=1 ahead o the otorcycle Then the acceleration o the otorcycle can be ound ro the equation = da yielding ( ) ( s -1 ) a = = = s - d ()(1 ) or twice the gravitational acceleration Note the inal velocity o the otorcycle is v (t )= 4 s -1 It s really oving along! 1 6

7 The tie it takes to overtake the car is then t = = )(1 s = 1 s - d / a ( )/( ) This is not an unreasonable nuber or the tie Notice that a double negative is appropriate here because when we do these types o calculations, the best we can say is that our result does not contradict our experience Dierent Approaches to solving the Proble: I ound that y two conditions or when the otorcycle overtakes the car sipliies to 1 at = d at = These equations look suggestive, so can I ind a nice physical reason or this equation? Suppose I ove with the car Then the car has zero velocity and the otorcycle has zero initial velocity, v =, but the acceleration o the otorcycle relative to the car is still the sae So in this reerence rae, the position o the otorcycle is and the velocity is The velocity o the otorcycle at t is Thus the velocity equation is just x 1 ( t )= 1 at v x,1 () t = a t v x,1 ( t ) = = a t So the reason the two conditions have such a siple or is that they are the result o solving the proble in a reerence rae oving with the car 7