Particle Motion on a Line v , I , • • 3 -_.l. __ -J.__ ••• _-tr--_J.. __ .J L__ .J_ .• I • I • • I • I • 2 --4 ~--~---~--~-. • I • • I , _"' ~__ ,J_. I I I I I I __~--~--~--~--+---~~p-~I--~I--~t o -I 1 •I 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 the function whose graph is shown above.,~ 1. At what value of t does the bug change direction? a. 2 b. 4 c. 6 d. 7 e. 8 2. What is the total distance the bug traveled from t =0 to t -8 ? a. 14 b. 13 c. 11 d. 8 e. 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 of 1 is the velocity of the particle zero? a. I b. 2 c. 3 d. 4 e. 5 4. The maximum acceleration attained on the interval 0 S; t S; 3 by the particle whose velocity is given by V(I) = 1 3 - 3/ 2 + 121+ 4 is a. 9 b. 12 c. 14 d. 21 e. 40
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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
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?
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 ?
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.
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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