Chapter 10 Velocity, Acceleration, and Calculus

Transcription

1 Chapter 10 Velocity, Acceleration, and Calculu The firt derivative of poition i velocity, and the econd derivative i acceleration. Thee derivative can be viewed in four way: phyically, numerically, ymbolically, and graphically. The idea of velocity and acceleration are familiar in everyday experience, but now we want you to connect them with calculu. We have dicued everal cae of thi idea already. For example, recall the following (retated) Exercie car from Chapter 9. Example 10.1 Over the River and Through the Wood We want you to ketch a graph of the ditance traveled a a function of elaped time on your next trip to viit Grandmother. Make a qualitative rough ketch of a graph of the ditance traveled,, a a function of time, t, on the following hypothetical trip. You travel a total of 100 mile in 2 hour. Mot of the trip i on rural intertate highway at the 65 mph peed limit. You tart from your houe at ret and gradually increae your peed to 25 mph, low down and top at a top ign. You peed up again to 25 mph, travel for a while and enter the intertate. At the end of the trip you exit and low to 25 mph, top at a top ign,and proceed to your final detination. The correct qualitative hape of the graph mean thing like not crahing into Grandmother garage at 50 mph. If the end of your graph look like the one on the left in Figure 10.1:1, you have eriou damage. Notice that Leftie graph i a traight line, the rate of change i contant. He travel 100 mile in 2 hour, o that rate i 50 mph. Imagine Grandmother urprie a he arrive! t t Figure 10.1:1: Leftie and Rightie go to Grandmother The graph on the right low to a top at Grandmother, but Rightie went though all the top ign. How could the police convict her uing jut the graph? She paed the top ign 3 minute before the end of her trip, 2 hour le 3 minute = 2-3/60 = 1.95 hr. Graph of her ditance for hort time interval around t =1.95 look like Figure 10.1:2 218

2 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS t t Figure 10.1:2: Two view of Rightie moving violation Wow, the mallet cale graph look linear? Why i that? Oh, yeah, microcope. What peed will the cop ay Rightie wa going when he paed through the interection? =7 mph He could even keep up with her on foot to give her the ticket at Grandmother. At leat Rightie willnothavetogotojaillikeleftiedid. You hould undertand the function verion of thi calculation: [1.955] [1.945] = = [t + t] [t] t = = [ ] [1.945] ( ) Exercie Set Look up your olution to Exercie or reolveit. Beuretoincludethefeatureof topping at top ign and at Grandmother houe in your graph. How do the peed of 65 mph and 25 mph appear on your olution? Be epecially careful with the lope and hape of your graph. We want to connect lope and peed and bend and acceleration later in the chapter and will ak you to refer to your olution. 2. A very mall-cale plot of ditance traveled v. time will appear traight becaue thi i a magnified graph of a mooth function. What feature of thi traight line repreent the peed? In particular, how fat i the peron going at t =0.5 for the graph in Figure 10.1:3? What feature of the large-cale graph doe thi repreent?

3 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS t t Figure 10.1:3: A microcopic view of ditance Velocity and the Firt Derivative Phyicit make an important ditinction between peed and velocity. A peeding train whoe peed i 75 mph i one thing, and a peeding train whoe velocity i 75 mph on a vector aimed directly at you i the other. Velocity i peed plu direction, while peed i only the intantaneou time rate of change of ditance traveled. When an object move along a line, there are only two direction, o velocity can imply be repreented by peed with a ign, + or. 3. An object move along a traight line uch a a traight level railroad track. Suppoe the time i denoted t, witht =0when the train leave the tation. Let repreent the ditance the train ha traveled. The variable i a function of t, = [t]. Weneeoetunitandadirection. Why? Explain in your own word why the derivative d What doe a negative value of d repreent the intantaneou velocity of the object. mean? Could thi happen? How doe the train get back? 4. Krazy Kouin Keith drove to Grandmother, and the reading on hi odometer i graphed in Figure 10.1:4. What wa he doing at time t =0.7? (HINT: The only way to make my odometer read le i to back up. He mut have forgotten omething.)

4 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS t Figure 10.1:4: Keith regreion 5. Portion of a trip to Grandmother look like the next two graph t t Figure 10.1:5: Poitive and negative acceleration Which one i ga, and which one i brake? Sketch two tangent line on each of thee graph and etimate the peed at thee point of tangency. That i, which one how lowing down and which peeding up? 10.2 Acceleration Acceleration i the phyical term for peeding up your peed... Your car accelerate when you increae your peed. Since peed i the firt derivative of poition and the derivative of peed tell how it peed up. In other word, the econd derivative of poition meaure how peed peed up... We want to undertand thi more clearly. The firt exercie at the end of thi ection ak you to compare the ymbolic firt and econd derivative with your graphical trip to Grandmother. A numerical approach to acceleration i explained in the following example. You hould undertand velocity and acceleration numerically, graphically, and ymbolically.

5 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 222 Example 10.2 The Fallen Tourit Reviited Recall the tourit of Problem 4.2. He threw hi camera and glae off the Leaning Tower of Pia in order to confirm Galileo Law of Gravity. The Italian police videotaped hi crime and recorded the following information: t = time (econd) = ditance fallen (meter) We want to compute the average peed of the falling object during each econd, from 0 to 1, from 1 to 2, and from 2 to 3? For example, at t =1, the ditance fallen i =4.8 and at t =2, the ditance i =18.5, o the change in ditance i =13.7 while the change in time i 1. Therefore, the average peed from 1 to 2 i 13.7 m/ec, Average peed = change in ditance change in time Time interval Average peed = t [0, 1] v 1 = = [1, 2] v 2 = = [2, 3] v 3 = = Example 10.3 The Speed Speed Up

6 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 223 Thee average peed increae with increaing time. How much doe the peed peed up during thee interval? (Thi i not very clear language, i it? How hould we ay, the peed peed up?) Interval to interval Rate of change in peed [0, 1] to [1, 2] a 1 = = 9.8?? [1, 2] to [2, 3] a 2 = = 9.8?? The econd peed peed up 9.8 m/ec during the time difference between the meaurement of the firt and econd average peed, but how hould we meaure that time difference ince the peed are average and not at a pecific time? The tourit camera fall continuouly. The data above only repreent a few pecific point on a graph of ditance v. time. Figure 10.2:6 how continuou graph of time v. height and time v. = ditance fallen. 40 h t t Figure 10.2:6: Continuou fall of the camera The computation of the Example 10.2 find [1] [0], [2] [1], and[3] [2]. Which continuou velocitie do thee bet approximate? The anwer i v[ 1 2 ], v[ 3 2 ],andv[ 5 2 ] -thetimeat the midpoint of the time interval. Sketch the tangent at time 1.5 on the graph of v. t and compare that to the egment connecting the point on the curve at time 1 and time 2. In general, the ymmetric difference f[x + δx 2 ] f[x δx 2 ] f 0 (x) δx give the bet numerical approximation to the derivative of y = f[x] when we only have data for f. The difference quotient i bet a an approximation at the midpoint. The project on Taylor Formula how algebraically and graphically what i happening. Graphically, if the curve bend up, a ecant to the right i too teep and a ecant to the left i not teep enough. The average of one lope below and one above i a better approximation of the lope of

7 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 224 Figure 10.2:7: [f (x + δx) f (x)] /δx v. [f (x + δx) f (x δx)] /δx the tangent. The average lope given by the ymmetric ecant, even though that ecant doe not pa through (x, f[x]). The general figure look like Figure 10.2:7. The bet time to aociate to our average peed in comparion to the continuou real fall are the midpoint time: Time Speed = t 0.50 v 1 = v[0.50] = v 2 = v[1.50] = v 3 = v[2.50] = 24.5 Thi interpretation give u a clear time difference to ue in computing the rate of increae in the acceleration: Time Rate of change in peed Ave[0.50&1.50] = a[1] = =9.8 Ave[1.50&2.50] = a[2] = =9.8 We ummarize the whole calculation by writing the difference quotient in a table oppoite the variou midpoint time a follow:

8 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 225 Firt and Second Difference of Poition Data Time Poition Velocity Acceleration Table 10.1: One-econd poition, velocity, and acceleration data Exercie Set 10.2 d The firt exercie eek your everyday interpretation of the poitive and negative ign of and d2 on the hypothetical trip from Example We need to undertand the mechanical interpretation of thee derivative a well a their graphical 2 interpretation. 1. Look up your old olution to Exercie or Example 10.1 and add a graphing table like the one from the Chapter 9 with lope and bending. Fill in the part of the table correponding to d and d2 uing the microcopic lope and mile and frown icon including + and ign. 2 Remember that d2 i the derivative of the function d 2 ; o, for example, when it i poitive, the function v[t] = d increae, and when it i negative, the velocity decreae. We alo need d to connect the ign of 2 with phyic and the graph of [t]. Ue your olution graph of time, 2 t, v. ditance,, to analyze the following quetion. (a) Where i your peed increaing? Decreaing? Zero? If peed i increaing, what geometric hape mut that portion of the graph of [t] have? (The graph of v[t] ha upward lope and poitive derivative, dv = d2 > 0, but we are aking how increae in v[t] = d 2 affect the graph of [t].) (b) I d ever negative in your example? Could it be negative on omeone olution? Why doe thi mean that you are backing up? (c) Summarize both the mechanic and geometrical meaning of the ign of the econd derivative d2 in a few word. When d2 i poitive...when d2 i negative (d) Why mut d2 be negative omewhere on everyone olution? 2 There are more accurate data for the fall of the camera in half-econd time tep:

9 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 226 Accurate Poition Data Time Poition Velocity Acceleration Table 10.2: Half-econd poition data 2. Numerical Acceleration Compute the average peed correponding to the poition in Table 10.2 above and write them next to the correct midpoint time o that they correpond to continuou velocitie at thoe time. Then ue your velocitie to compute acceleration at the proper time. Simply fill in the place where the quetion mark appear in the velocity and acceleration Table 10.3 and 10.4 following thi exercie. The data are alo contained in the Gravity program o you can complete thi arithmetic with the computer in Exercie HINTS: We begin the computation of the acceleration a follow. Firt, add midpoint time to the table and form the difference quotient of poition change over time change: Difference of the Half-Second Poition Data Time Poition Velocity Acceleration = = ??? =? ? ? ? Table 10.3: Half-econd velocity difference Next, form the difference quotient of velocity change over time change:

10 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 227 Second Difference of the Half-Second Poition Data Time Poition Velocity Acceleration = ?? ? =? ? ? ? Table 10.4: Half-econd acceleration difference 10.3 Galileo Law of Gravity The acceleration due to gravity i a univeral contant, d2 2 = g. Data for a lead cannon ball dropped off atallcliff are contained the the computer program Gravity. The program contain time-ditance pair for t =0, t =0.5, t =1.0,, t =9.5, t =10. A graph of the data i included in Figure 10.3: h t t Figure 10.3:8: Free fall without airfriction

11 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 228 Galileo famou obervation turn out to be even impler than the firt conjecture of a linear peed law (which you will reject in an exercie below). He found that a long a air friction can be neglected, the rate of increae of peed i contant. Mot triking, the contant i univeral - it doe not depend on the weight of the object. Galileo Law of Gravity The acceleration due to gravity i a contant, g, independent of the object. The value of g depend on the unit of time and ditance, 9.8m/ec 2 or 32ft/ec 2. Fluency in calculu mean that you can expre Galileo Law with the differential equation d 2 2 = g The firt exercie for thi ection ak you to clearly expre the law with calculu. Exercie Set Galileo Law and d2 2 Write Galileo Law, The rate of increae in the peed of a falling body i contant. in term of the derivative of the ditance function [t]. What derivative give the peed? What derivative give the rate at which the peed increae? We want you to verify Galileo obervation for the lead ball data in the Gravity program. 2. Numerical Gravitation (a) Ue the computer to make lit and graph of the peed from 0 to 1 2 econd, from 1 2 to 1, and o forth, uing the data of the Gravity program. Are the peed contant? Should they be? (b) Alo ue the computer to compute the rate of change in peed. Are thee contant? What doe Galileo Law ay about them? d 3. Galileo Law and the Graph of Galileo Law i eaiet to confirm with the data of the Gravity program by looking at the d graph of (becaue error meaurement are magnified each time we take difference of our data). What feature of the graph of velocity i equivalent to Galileo Law? In the Problem 4.2 you formulated a model for the ditance an object ha fallen. You oberved that the farther an object fall, the fater it goe. The implet uch relationhip ay, The peed i proportional to the ditance fallen. Thi i a reaonable firt gue, but it i not correct. We want you to ee why. (Compare the next problem with Problem 8.5.)

12 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS 229 Problem 10.1 Try to find a contant to make the conjecture of Problem 4.2 match the data in the program Gravity, that i, make the differential equation d = k predict the poition of the falling object. Thi will not work, but trying will how why. There are everal way to approach thi problem. You could work firt from the data. Compute the peed between 0 and 1 2 econd, between 1 2 and 1 econd, and o forth, then divide thee number by and ee if the lit i approximately the ame contant. The differential equation d = k ay it hould be, becaue v = d = k v = k i contant. There i ome help in the Gravity program getting the computer to compute difference of the lit. Remember that the time difference are 1 2 econd each. You need to add a computation to divide the peed by, [t ] [t] 1 d at t each divided by [t]. Notice that there i ome computation error caued by our approximation to d actually being bet at t but only having data for [t]. Be careful manipulating the lit with Mathematica becaue the velocity lit ha one more term than the acceleration lit. Another approach to rejecting Galileo firt conjecture i to tart with the differential equation. We can olve d = k with the initial [0] = 0 by method of Chapter 8, obtaining [t] =S 0 e kt. What i the contant S 0 if [0] = 0. How do you compute [0.01] from thi? See Bug Bunny Law of Gravity, Problem 8.5 and Exercie Set 8.2. The zero point caue a difficulty a the preceding part of thi problem how. Let ignore that for the moment. If the data actually are a olution to the differential equation, = S 0 e kt,then Log[] =Log[S 0 e kt ]=Log[S 0 ]+Log[e kt ]=σ 0 + kt o the logarithm of the poition (after zero) hould be linear. Compute the log of the lit of (non-zero) poition with the computer and plot them. Are they linear? 10.4 Project Several Scientific Project go beyond thi baic chapter by uing Newton far-reaching extenion of Galileo Law. Newton Law ay F = m a, the total applied force equal ma time acceleration. Thi allow u to find the motion of object that are ubjected to everal force.

13 Chapter 10 - VELOCITY, ACCELERATION and CALCULUS The Falling Ladder Example 7.14 introduce a imple mathematical model for a ladder liding down a wall. The rate at which the tip reting againt the wall lide tend to infinity a the tip approache the floor. Could a real ladder tip break the ound barrier? The peed of light? Of coure not. That model neglect the phyical mechanim that make the ladder fall - Galileo Law of Gravity. The project on the ladder ak you to correct the phyic of the falling ladder model Linear Air Reitance A feather doe not fall off atallcliff a fat a a bowling ball doe. The acceleration due to gravity i the ame, but air reitance play a ignificant role in counteracting gravity for a large, light object. A baic project on Air Reitance explore the path of a wooden ball thrown off the ame cliff a the lead ball we jut tudied in thi chapter Bungee Diving and Nonlinear Air Reitance Human bodie falling long ditance are ubject to air reitance, in fact, ky jumper do not keep accelerating but reach a terminal velocity. Bungee jumper leap off tall place with a big elatic band hooked to their leg. Gravity and air reitance act on the jumper in hi initial flight, but once he reache the length of the cord, it pull up by an amount depending on how far it i tretched. The Bungee Jumping Project ha you combine all thee force to find out if a jumper hit the bottom of a canyon or not The Mean Value Math Police The police find out that you drove from your houe to Grandmother, a ditance of 100 mile in 1.5 hour. How do they know you exceeded the maximum peed limit of 65 mph? The Mean Value Theorem Project anwer thi quetion Symmetric Difference The Taylor Formula (from the project of that name) how you why the bet time for the velocity approximated by ([t 2 ] [t 1 ])/(t 2 t 1 ) i at the midpoint, v[(t 2 +t 1 )/2]. Thi i a general numerical reult that you hould ue any time that you need to etimate a derivative from data. The project how you why.