Section 2.3: Particle Motion

Transcription

1 : Particle Motion Particle Motion Particle motion describes the physics of an object (a point) that moves along a line; usually horizontal There are three different functions that model this action Position! ", $ " Determines where the particle is located on the! axis at a given time " Velocity & " = $ " Determines how fast the position is changing at a time " and direction of movement Acceleration ) * = & " = $ " Determines how fast the velocity is changing at a time "; sign indicates if the velocity is increasing or decreasing AP Calculus 1

2 What do the following statements indicate? If the acceleration is positive, then the velocity is increasing For the particle to change direction velocity must change signs At rest Means velocity '(!) = # If the velocity is positive, then the particle is moving to the right If the acceleration is negative, then the velocity is decreasing Initially Means time! = # If the velocity is negative, then the particle is moving to the left At the origin Means position x(!) = %(!) = # What do the following statements indicate? The particle is speeding up when velocity and acceleration have the same signs The particle is slowing down when velocity and acceleration have different signs AP Calculus 2

3 Ex 1) The data in the table below give selected values for the velocity, in meters min, of a particle moving along the ) axis. The velocity +(-) is a differentiable function of time -. time / (min) velocity 6(/) (meters/min) (a) At / = ;, is the particle moving to the right or to the left? Since the velocity is negative at / = ;, the particle is moving to the left. (b) Is there a time on the interval ; / =>?@AB when the particle is at rest? By the Since velocity is Since velocity goes from Intermediate Value differentiable, it negative to positive between / = ; & / = >, it must pass Theorem, F / must be continuous. through zero which means such that the the particle is at rest. particle is at rest. Ex 1) The data in the table below give selected values for the velocity, in meters min, of a particle moving along the ) axis. The velocity +(-) is a differentiable function of time -. time / (min) velocity 6(/) (meters/min) (c) Use data from the table to find an approximation for : (<=) and explain the meaning of : (<=) in terms of the motion of the particle. Show the computations and indicate units of measure. >?@ABC@D AE?@CE@CEDFG? CHHDIDJC@AFE G?AEK C6DJCKD CHHDIDJC@AFE FE AE@DJ6CI L, <N. Q = : / = S T <N L = N U = < N B V BAE N : (<=) represents the acceleration of the particle at / = <= minutes. AP Calculus 3

4 (a) At 3 = 4 seconds, is the particle moving to the right or left? Since velocity is positive at 3 = 4 (5(3) is above the 6 axis), particle is moving to the right. (b) Over what time interval is the particle moving to the left? 89:;: <= 5 3 A:B@8 6 CD<= Particle is moving to the left on the interval E < 3 G since the velocity is negative (c) At 3 = 4 seconds, is the acceleration of the particle positive or negative? 89 : ; (3) < = >? : ; 3 > = AB 3 = 4? Since slope of velocity is negative at 3 = 4, acceleration of particle is negative. (d) What is the average acceleration of the particle over the interval 5 3 4? Show the computations and indicate units of measure. E = : 3 = F G 4 5 = H 5 E H JB 5 I 9KL 5 AP Calculus 4

5 (e) At what time 3 in the given interval is the particle farthest to the right? From 3 = = 8, velocity of the particle is positive. From 3 = = 8, the particle is moving to the right. At 3 = 8, velocity of the particle changes from positive to negative. At 3 = 8, the particle is begins moving to the left. :6 3 = 8 the particle is farthest to the right. (g) On what interval of time 3 is the particle speeding up? On what interval of time 3 is the particle slowing down? Speeding up when velocity & acceleration have the same signs Velocity & acceleration are both positive on 5, 7 and 8, 9. Particle speeding up on time intervals 5, 7 and 8, 9. AP Calculus 5

6 (g) On what interval of time 3 is the particle speeding up? On what interval of time 3 is the particle slowing down? Slowing down when velocity & acceleration have different signs There are no time intervals where velocity is negative and acceleration is positive. Velocity is positive and acceleration is negative on time interval 6, 8. Particle slowing down on time interval 6, 8. Ex 3) A particle moves along the! axis so that at time # its position is given by s # = # & 6# ( + 9# (a) At # = 0, is the particle moving to the right or to the left?./ 0(2) < > 2? Find : ; 9 : 0 9 = : ; 9 = <9 = >=9 +? Evaluate : ; 2 : 0 2 = : ; 2 = <(2) = >=(2) +? 0 2 =? Since 0 2 > 2, the particle is moving to the right. (b) At # = 1, is the velocity of the particle increasing or decreasing?./ 0 (>) < > > 2? Find : ;; 9 : 0 9 = : ;; 9 = A9 >= 0 > = : ;; > = A(>) >= 0 ; (>) = A Since 0 ; (>) < 2, the velocity of the particle is decreasing. AP Calculus 6

7 Ex 3) A particle moves along the! axis so that at time # its position is given by s # = # & 6# ( + 9# (c) Find all values of # for which the particle is moving to the left. -./0/ 12 3(5) < 8? Find : ; 5 : 3 5 = : ; 5 = <5 = >=5 +? Find 5 where 3(5) < 8: <5 = >=5 +? < 8 < 5 + < < 8 Particle is moving to the left on the interval >, < < 5 > 5 < < 8 since the velocity is negative Test 5 = > 5 = < intervals + + > < 3(5) < 8 when >, < AP Calculus 7