3024 Rectilinear Motion On a line AP Calculus

3024 Rectilinear Motion On a line AP Calculus

Position Defn: Rectilinear Motion: Movement of object in either direction along a coordinate line (x-axis , or y-axis) s sic y ph s(t) Spotia = position function - position versus time graph (historical note: Latin h at m x(t) = horizontal axis y(t) = vertical axis a directed distance (a vector quantity) of the particle from some point, Direction and quantity p, at instant t. negative time = time before s(t) Located to the right positive – the particle is____________________ Located to the left negative – the particle is ____________________ = 0 At the origin – the particle is ____________________ _

Velocity v(t) = velocity function - the rate of change of position Velocity gives both quantity of change and direction of change (again a vector quantity) Speed finds quantity only. - absolute value of Magnitude only velocity (a scalar quantity) Rem: Average Velocity = change in position over change in time = Average speed = Instantaneous Velocity the derivative Ticketed speed

Velocity v(t) = velocity function - the rate of change of position = Instantaneous Velocity the derivative v(t) increasing positive – the particle’s position is __________ < velocity in a positive direction - ____________ Moving to the right decreasing negative – the particle’s position is ___________ < velocity in a negative direction - ____________ Moving to the left stationary not moving = 0 - the particle is _______________ * {This is the 1 st Derivative Test for increasing /decreasing!}

Acceleration a(t) = acceleration function - rate of change of velocity a(t) increasing positive - velocity is _________________ < acc. in a positive direction – _____________ Pushed to the right decreasing negative - velocity is _________________ Pushed to the left < acc. in neg. direction – _______________ constant cruise control = 0 - velocity is _________________ {This is the 2 nd derivative test for concavity} CAREFUL: This is not SPEEDING UP or SLOWING DOWN!

Speed and Direction Determining changes in Speed speed increasing if v(t) and a(t) have same sign - Moving Pushed also for v(t) = 0 and a(t) 0 speed decreasing if v(t) and a(t) Moving have opposite signs - Pushed Or Moving Pushed Determining changes in Direction and a(t) 0 Ball bouncing no change if both v(t) = 0 and a(t) = 0 Sitting still direction changes if v(t) = 0

Method (General): t= 0 Se 1) Find the Critical Numbers in First and Second Derivatives. 1) Answer any questions at specific locations. 2) Do the Number Line Analysis (Brick Wall). 1) Find direction - moving , pushed , and speed 3) Identify the Change of Direction locations 1) Find values at beginning, ending, and change of direction times. 4) Sketch the Schematic graph. 5) Find the Displacement and Total Distance Traveled.

Example: A particle’s position on the y –axis is given by: 1) Find y(t), v(t) and a(t) at t = 2. Interpret each value. Located 14 units left Moving right 8 units/sec Pushed 36 units 2/sec Speed is increasing

Example: A particle’s position on the y –axis is given by: Number line analysis 21 2) Determine the motion within each interval: location, direction moving and direction pushed. Moving left v(t) pos neg a(t) m 6 P s -3 -2 3) Find the values at t = -3, t -1 1 alues 0 ion y v t a u q e inal =O 3 rigand where the particle changes directions. 4) Find the Displacement and Total Distance Traveled. Ending – beginning 21 -21=0 36+9+9+36 2 3 >36 > 9 >9 >36 -15

Example 2 : A particle’s position on the x –axis is given by: Find and interpret x(t), v(t), and a(t) at t = 5

Example: A particle’s position on the x –axis is given by: v a m p s Sketch:

v a m p s

v a m p s

v a m p s

v a m p s

v a m p s

v a m p s

v a m p s

Last Update • 11/22/10 • Assignment: work sheet - Swokowski