3024 Rectilinear Motion AP Calculus On a line

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

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 velocity (a scalar quantity) Rem: Average Velocity = change in position over change in time = = Instantaneous Velocity  the derivative Magnitude only Average speed Ticketed speed

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

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

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

Method (General): 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. Set = 0

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: 2) Determine the motion within each interval: location, direction moving and direction pushed. 3) Find the values at t = -3, t = 3 and where the particle changes directions. 4) Find the Displacement and Total Distance Traveled. v(t) a(t) mPsmPs neg pos neg Moving left > 36 > 9 > 36 >9>9 Original equation y values Ending – beginning 21-21=0 Number line analysis

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: Sketch: v a mpsmps

v a mpsmps

v a mpsmps

v a mpsmps

v a mpsmps

v a mpsmps

v a mpsmps

v a mpsmps

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