Particle Motion (AKA Rectilinear Motion)

Vocabulary Rectilinear Motion –Position function –Velocity function Instantaneous rate of change (position  time) –Speed function Absolute value of velocity –Acceleration Function Instantaneous rate of change (velocity  time) Speeding up/Slowing down

Particle Motion Motion on a line Moving in a positive 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 time position Moving in a negative direction from the origin Position function: s(t) s = position (sposition duh!) t = time s(t)= position changes as time changes

Example Use the position and time graph to describe how the puppy was moving time position

Velocity Rate –position change vs time change –Velocity can be positive or negative positive: going in a positive direction negative: going in a negative direction Velocity Position

Velocity Rate at which a coordinate of a particle changes with time s(t) = position with respect to time Instantaneous velocity at time t is: time position v(t) = positive – increasing slope – moving in a positive direction v(t) = negative – decreasing slope – moving in a negative direction

Velocity function Velocity is the slope of the position function (change in position /change in time) velocity = – This is instantaneous rate of change (position  time) PositionVelocityMeaning Positive SlopePositive y’smoving in a positive direction Negative slopeNegative y’sMoving in a negative direction

Practice Let s(t)= t 3 -6t 2 be the position function of a particle moving along an s-axis were s is in meters and t is in seconds. –Graph the position function –On a number line, trace the path that the particle took. –Where will the velocity be positive? Negative? –Graph the velocity function –Identify on the velocity function when the particle was heading in a positive direction and when it was heading in a negative direction.

Example: s(t)= t 3 -6t 2 position time velocity time speed

Velocity vs Speed Speed is 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 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 directions but both have a speed of |v|=5 m/s

Practice Graph the velocity function What 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 acceleration of the particle at time t is: **The second derivative of the position function!!

Example Let s(t) = t 3 – 6t 2 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

Speeding Up & Slowing Down Speeding up velocity and acceleration are the same sign. Slowing down when velocity and acceleration are opposite signs.

Example When is s(t) speeding up and slowing down? position time velocity acceleration

Velocity & Acceleration Functions Slowing down Velocity + Acceleration - Speeding up Velocity - Acceleration - Slowing down Velocity - Acceleration + Speeding up Velocity + Acceleration +

Analyzing Motion GraphicallyAlgebraicallyMeaning Position Velocity  Acceleration 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/stopped 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) = 2t 3 -21t 2 +60t+3 Analyze the motion of the particle for t>0 GraphicallyAlgebraicallyMeaning  Position Velocity Acceleration Never 0 (t>0), always postive Always on postive side of number line <t<2 going pos direction t=2 turning 2<t<5 going neg. direction t=5 turning t>5 going pos. direction t=0 t=2 t= v a <t<2 slowing down 2<t<3.5 speeding up 3.5<t<5 slowing down 5<t speeding up

Example Suppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t 3 -21t 2 +60t+3 Analyze the motion of the particle for t>0 position velocity Acceleration

position velocity Acceleration Position t Direction of motion stop positive direction negative direction positive direction 02 5 v(t) /2 a(t) slowing down speeding up slowing down speeding up

Applications: Gravity s = position (height) s 0 = initial height v 0 = initial velocity t = time g= acceleration due to gravity –g=9.8 m/s 2 (meters and seconds) –g=32 ft/s 2 (feet and seconds) s0s0

Applications: Gravity at time t= 0 an object at a height s 0 above the Earth’s surface is given an upward or downward velocity of v 0 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 the height s=s(t) of the object is : – g= acceleration due to gravity –g=9.8 m/s 2 (meters and seconds) –g=32 ft/s 2 (feet and seconds) s axis s0s0

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)= feet