Calculus.ppt

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1、Calculus,6.3 Rectilinear Motion,6.3 Rectilinear Motion,Vocabulary,Day 1: Rectilinear Motion Position function Velocity function Instantaneous velocity Day 2: Speed function Instantaneous speed Day 3 Acceleration Function Speeding up Slowing down,Rectilinear Motion,Motion on a line,Moving in a positi

2、ve direction from the origin,Moving in a negative direction from the origin,Position Function,Horizontal axis: time Vertical Axis: position on a line,Moving in a positive direction from the origin,Moving in a negative direction from the origin,Position function: s(t) s = position (sposition duh!) t

3、= time s(t)= position changes as time changes,Sketchpad Example,Example,Use the position and time graph to describe how the puppy was moving,Velocity,Rate position change vs time change Velocity can be positive or negative positive: going in a positive direction negative: going in a negative directi

4、on,Velocity,Position,Velocity function,Velocity is the slope of the position function (change in position /change in time) velocity =Technically this is instantaneous velocity,Velocity,Rate at which a coordinate of a particle changes with time Insanities velocity s(t) = position with respect to time

5、 Instantaneous velocity at time t is:,v(t) = positive increasing slope moving in a positive direction,v(t) = negative decreasing slope moving in a negative direction,Practice,Let s(t)= t3-6t2 be the position function of a particle moving along an s-axis were s is in meters and t is in seconds. Graph

6、 the position function On a number line, trace the path that the particle took. Where will the velocity be positive? Negative? Graph the instantaneous velocity. Identify on the velocity function when the particle was heading in a positive direction and when it was heading in a negative direction.,Ve

7、locity or Speed,Speed change in position with respect to time in any direction Velocity is the change in position with respect to time in a particular direction Thus Speed cannot be negative because going backwards or forwards is just a distance Thus Velocity can be negative because we care if we go

8、 backwards,Speed,Absolute Value of Velocity,example: if two particles are moving on the same coordinate line with velocity of v=5 m/s and v=-5 m/s,then they are going in opposite directionsbut both have a speed of |v|=5 m/s,Example - s(t)= t3-6t2,time,speed,Practice,Graph the velocity function What

9、will the speed function look like? At what time(s) was the particle heading in a negative direction? Positive direction?,Acceleration,the rate at which the velocity of a particle changes with respect to time. If s(t) is the position function of a particle moving on a coordinate line, then the instan

10、taneous acceleration of the particle at time t isor,Example,Let s(t) = t3 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the instantaneous acceleration a(t) and shows the graph of acceleration verses time,Day 3: Speeding Up & Slowing

11、down,Speeding up when slope of speed is positive Slowing down when slope of speed is negative,Example,When is s(t) speeding up and slowing down?,speed,acceleration,Velocity & Acceleration function,Slowing down,Velocity +,Acceleration -,Speeding up,Velocity -,Acceleration -,Slowing down,Velocity -,Ac

12、celeration +,Speeding up,Velocity +,Acceleration +,Analyzing Motion,Positive “s” values,Positive side of the number line,Negative side of the number line,Negative “s” values,s(t)=velocity.,Look for Critical Pts,Postive “v” values,0 “v” values (CP),Negative “v” values,Moving in + direction,Turning/st

13、opped,Moving in a direction,v(t)=acceleration,Look for Critical Pts,+ a, + v = speeding up - a, - v = speeding up + a, - v = slowing down - a, + v = slowing down,Example,Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3-21t2+60t+3 Analyze the motion

14、of the particle for t0,Position,Velocity,Acceleration,Never 0 (t0), always postive,Always on postive side of number line,+,-,+,0,0,0t2 going pos direction,t=2 turning,2t5 going neg. direction,t=5 turning,t5 going pos. direction,t=0,t=2,t=5,+,-,-,+,-,-,+,+,0t2 slowing down,2t3.5 speeding up,3.5t5 slo

15、wing down,5t speeding up,Example,Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3-21t2+60t+3 Analyze the motion of the particle for t0,position,velocity,Acceleration,position,velocity,Acceleration,Position,Direction of motion,+,-,+,stop,stop,positiv

16、e direction,negative direction,positive direction,v(t),+,-,+,7/2,a(t),-,+,slowing down,speedingup,slowing down,speedingup,Day 4: Applications; Gravity,s = position (height) s0= initial height v0= initial velocity t = time g= acceleration due to gravity g=9.8 m/s2 (meters and seconds) g=32 ft/s2 (fee

17、t and seconds),s0,Day 4: Applications; Gravity,at time t= 0 an object at a height s0 above the Earths surface is given an upward or downward velocity of v0 and moves vertically (up or down) due to gravity. If the positive direction is up and the origin is the surface of the earth, then at any time t

18、 the height s=s(t) of the object is :g= acceleration due to gravity g=9.8 m/s2 (meters and seconds) g=32 ft/s2 (feet and seconds),s axis,s0,Example,Nolan Ryan was capable of throwing a baseball at 150ft/s (more than 102 miles/hour). Could Nolan Ryan have hit the 208 ft ceiling of the Houston Astrodome if he were capable of giving the baseball an upward velocity of 100 ft/s from a height of 7 ft?,the maximum height occurs when velocity = 0,t=100/32=25/8 seconds,s(25/8)=163.25 feet,