AP Calculus AB / BC Particle Motion 1. A particle moves along the x-axis so that its velocity at any given time is 푣(푡) = 2푡 − 3 and 푥(0) = 3.
a.) Find the equation for position of the particle as a function of t, 푥(푡).
b.) Find the position of the particle at t = 3.
c.) At what values of t does the particle change direction?
d.) Where is the particle when the position is a minimum?
2. A particle moves along the x-axis so that its position at any time 푡 ≥ 0 is given by 푥(푡) = 푥3 − 6푥2 + 12푥 + 4. (a) Determine 푣(푡).
(b) Determine the interval(s) on which the particle is moving right.
(c) Determine 푎(푡).
(d) Determine the interval(s) on which the velocity of the particle is decreasing.
AP Calculus AB / BC Particle Motion 3. A particle moves along the x-axis so that its velocity at any time 푡 ≥ 0 is given by 푣(푡) = 3t2 − 16t + 5. The position 푥(푡) is 4 for 푡 = 2.
a.) Find a function that describes the position of the particle with respect to time. Show the analysis that leads to your conclusion.
b.) For t 0determine intervals on which the particle is moving left. Show the analysis that leads to your conclusion.
c.) Determine the acceleration of the particle at t =1s e c . Show your work.
d.) Find the maximum position of the particle on the interval form [0, 8]. Show the analysis that leads to your conclusion.
4. The acceleration of a particle moving on the x-axis is given by 푎(푡) = 6푡 − 18. At 푡 = 0, the velocity of the particle is 푣(0) = 24 and at 푡 = 1, its position is 푥(1) = 20.
(a) Write an expression for velocity in terms of t.
(b) Write an expression for position, 푥(푡), in terms of t.
(c) Determine the time(s), t, when the particle’s velocity is zero.
(d) Describe the motion of the particle from 푡 = 2 to 푡 = 4 seconds. Indicate both direction and distance.
AP Calculus AB / BC Particle Motion 5. Given the graph the velocity function of a particle moving on the x-axis where t is measured in seconds and vt() is the in inches per second, answer the following.
v(t)
− − time −
a. When is the particle moving right? Justify your answer. b. When is the particle moving left? Justify your answer. c. On the interval 03t when is the particle’s position a maximum? Justify your answer. d. On the interval 14t when is the particles position a minimum? Justify your answer. e. When is the particle’s acceleration positive? Justify your answer. f. When is the particle’s acceleration negative? Justify your answer. g. When is the particle speeding up? Justify your answer. h. When is the particle slowing down? Justify your answer.