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Particle Motion on a Line

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Questions 1 - 2 refer to the following situation.

A bug begins to crawl up a vertical wire at time t = O. The velocity v of the bug at time t, 0 S; (S; 8. is givei/by thefunction whose graph is shown above.,~

1. At what value of t does the bug change direction?

a. 2b. 4c. 6d. 7e. 8

2. What is the total distance the bug traveled from t =0 to t -8 ?

a. 14b. 13c. 11d. 8e. 6

3. A particle moves along the x-axis so that its position at time 1 is given by x(t) = (2 - 6/+ 5. For what value of1 is the velocity of the particle zero?

a. Ib. 2c. 3d. 4e. 5

4. The maximum acceleration attained on the interval 0 S; t S; 3 by the particle whose velocity is given by

V(I) = 13- 3/2 + 121+ 4 is

a. 9b. 12c. 14d. 21e. 40

5. A particle moves along the x-axis so that its acceleration at any time t is a(t) = 21-7. If the initial velocity of thepanicle is 6, at what time t during the interval 0 ~ t ~ 4 is the particle farthest to the right?

a. 0b. 1c. 2d. 3e. 4

6. A particle moves along the x-axis so that at time t 2 0 its position is given by x (t) = 2(3 - 21 (2 + 72t - 53. Atwhat time ( is the particle at rest?

a. 1= 1 onlyb. t == 3 only

7c. t=- only

2

d.7

I; 3 and t=-2

e. 1= 3 and 1=4"

t (see) 0 2 4 6')

5 2 8 3a (I) (ftlsec-)7.

The data for the acceleration a(t) of a car from 0 to 6 seconds are given in the table above. If the velocity at t=0 is 11 feet per second, the approximate value of the velocity at t = 6, computed using a left-hand Riemannsum with three subintervals of equal length, is

a. 26 ftlsecb. 30 ftlsecc. 37 ftlsecd. 39 ftlsece. 41 ftlsec

8. At time t e 0, the' acceleration ofa particle moving on the x-axis is a(t) = t+ sint. At t = 0, thevelocity of the particle is -2. For what value of t will the velocity of the particle be zero?

a. 1.02b. 1.48c. 1.85d. 2.81e. 3.14

9. Aparticle moves along the x-axis so that at any time t ~ 0, its velocity is given by v(t) = 3 +4.1 cos(0.9t).What is the acceleration of the particle at time t =4?

a. -2.016b. -0.677c. 1.633d. 1.814e. 2.978

10. The position of an object attached to a spring is given by yet) = i cos(5t) - ~ sin(5t), where tis

time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to O?I

a. Zerob. Threec. Fived. Sixe. Seven

11. A particle moves along the x-axis so that at any time t ~ 0, its velocity is given by v(t) = cos( 2 - t2). The

position of the particle is 3 at time t =0. What is the position of the particle when its velocity is first equal toO?

a. 0.411b. 1.310c. 2.816d. 3,091e. 3.411

3

12. The height h, in meters, of an object at time t is given by h(t) = 24t+ 24t2- 16t2

• What is the height of theobject at the instant when it reaches its maximum upward velocity?

a. 2.545 metersb. 10.263 metersc. 34.125 metersd. 54.889 meterse. 89.005 meters

2002 ApCPJCALCULUS AB FREE-RESPONSE QUESTIONS- .X z: 2./2-3. An object moves along the x·axis with initial position x(O) = 2. The velocity of the object at time t ~ 0

is given by v(t) =Sin( ~ t ).

(a) What is the acceleration of the object at time t = 4 ?

(b) Consider the following two statements.

Statement I: For 3 < t < 4.5, the velocity of the object is decreasing.Statement II: For 3 < t < 4.5, the speed of the object is increasing.

Are either or both of these statements correct? For each statement provide a reason why it is correct or notcorrect.

(c) What is the total distance traveled by the object over the time interval 0 S I S 4 ?

(d) What is the position of the object at time t = 4 ?

.;i

t 0 10 20 30 40 50 60 70 80(seconds)v(t) 5 14 22 29 35 40 44 47 49

(feet per second)

4. Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 :5 t :5 80 seconds, asshown in the table above.

(a) Find the average acceleration of rocket A over the time interval 0 :5 t :5 80 seconds. Indicate units ofmeasure.

J70

(b) Using correct units, explain the meaning of v(t) dt in terms ofthe rocket's flight. Use a midpoint10 •

J70

Riemann sum with 3 subintervals of equal length to approximate v(t) dt.10

(c) Rocket B is launched upward with an acceleration of a(t) = ~feet per second per second. At timevt + 1

t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which ofthe two rockets is traveling faster at time t = 80 seconds? Explain your answer.

Ap· CALCULUS AS 2002 SCORING GUIDEUNES

Question 3An object moves along the x..axis with initial position z (0) = 2. The velocity of the object at time t ~ 0 is given

by v(t) = sin ( it).(a) What is the acceleration of the object at time t = 4?(b) Consider the following two statements.

Statement I: For 3 < t < 4.5, the velocity of the object is decreasing.Statement II: For 3 < t < 4.5, the speed of the object is increasing,

Are either or both of these statements correct? For each statement provide a reason why it is correct or notcorrect.

(c) What is the total distance traveled by the object over the time interval 0 ~ t ~ 4?(d) What is the position of the object at time t = 4?

. i

(a) a(4) = v'(4) = ~cos(~)11'

= -'6 or -0.523 or -0.524

1: answer

(b) On 3 < t < 4.5: r I correct, with reasonaCt) = v'(t) = ~cos(it) < 0 3 1 ; II correctStatement I is correct since aCt) < O. 1: reason for IIStatement II is correct since v(t) < 0 andaCt) < O.

(c) Distance = J:4Iv(t)ldt = 2.387 limits of 0 and 4 on an integral. 0

OR of vet) or /v(t)/3 (11') 3 1 : orx(t) = --cos -t + - + 211' 3 11'

uses x(O) and x(4) to computex(O) = 29 distance

x(4) = 2 + 211' = 3.43239 31: handles change of direction at

vet) = 0 when t = 3 student's turning point6

x(3) = - + 2 = 3.90986 1: answer11'

15 0/1 if incorrect turning point orIX(3) - x(O)1 + Ix(4) - x(3)1 = 211' = 2.387no turning point

(d) x(4) = x(O) + 104v(t) dt = 3.432 2·1 ~ :

integral

answerOR OR

3 (11') 3x(t) = -~cos(it) + Cx( t) = - - cos - t + - + 2

r"11' 3 11'

9 2 1: answerx(4) = 2 + 211' = 3.4320/1 if no constant of integration

Copyright C 2002 by College Er!trame Examination Board. All rights reserved.Advanced Placement Prognun and AP are registered trademarks ofllK College Entrance Examination Board.

45

Ap® CALCULUS AB2006 SCORING GUIDELINES

Question 4

t0 10 20 30 40 50 60 70 80

(seconds)

v(t)5 14 22 29 35 40 44 47 49

(feet per second)

RocketA has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t ;;;;;0seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 ::; t ::;80 seconds, asshown in the table above.(a) Find the average acceleration of rocket A over the time interval 0::; t ::;80 seconds. Indicate units of

. imeasure.

r70(b). Using correct units, explain the meaning of JIO v( t) dt in terms of the rocket's flight. Use ~~idpoint

170Riemann sum with 3 subintervals of equal length to approximate v( t) dt.

10

(c) Rocket B is launched upward with an acceleration of a( t) = ~ feet per second per second. At time-u + 1

t = 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which ofthe two rockets is traveling faster at time t = 80 seconds? Explain your answer.

(a) Average acceleration of rocket A is 1 : answer

v(80) - v(O) _ 49 - 5 _ 11 ft/ 280 - 0 - 80 - 20 see

170(b) Since the velocity is positive, v( t) dt represents the

10

distance, in feet, traveled by rocket A from t = 10 secondsto t = 70 seconds.

A midpoint Riemann sum is20[v(20) + v( 40) + v(60)]= 20[22 + 35 + 44] = 2020 ft

(c) Let vB(t) be the velocity of rocket B at time t.

vB(t) = J ~ dt = 6~ + C-u + 1

2 = vB(O) = 6 + CvB(t) ;;;;;6.Jt+T - 4

vB(80) = 50 > 49 = v(80)

Rocket B is traveling faster at time t = 80 seconds.

Units of e] sec2 in (a) and ft. in (b)

{

I: explanation3: 1: uses v(20), v( 40), v(60)

1 : value

1 : 6.Jt+f1 : constant of integration

4 : 1: uses initial condition1 : finds vB(80), compares to v(80),

and draws a conclusion

1 : units in (a) and (b)

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