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13 
Kinematics
using Differentiation
What is distance, velocity, acceleration? Differentiate a distance function to get speed (velocity) function Differentiate a velocity function to get acceleration function Linking distance, speed, and acceleration functions using differentiation and introduce antidifferentiation 
Ex17.02 NuLake p98 
Bouncing ball & speed graph (java applet) Leaning tower of pizza (java applet) Rollercoaster: slow to load but well worth the wait Moving man java (great) 
Distance function S= 



Speed function V= 



Acceleration Function A= 
Find the acceleration of a car at t = 2 seconds, given the distance (metres) is defined by


Steps: 
d = t^{2}  6t +15 
1) Differentiate to find 'speed' function 
v = 2t  6 
2) Differentiate again to find the 'acceleration' function 
a = 2 
3) Substitute t = 4 into the 'acceleration' function 
Note: nowhere to substitue in t = 4 (acceleration here is always 2) 
4) Acceleration at t = 4 sec 
a = 2 m/s/s 
Find the acceleration of a car at t = 2 seconds, given the distance (m) is defined by d = 3t^{3}  9t^{2} + 18 

Steps: 
d = 3t^{3}  9t^{2} + 18 
1) Differentiate to find 'speed' function 
v = 9t^{2}  18t 
2) Differentiate again to find the 'acceleration' function 
a = 18t  18 
3) Substitute t = 2 into the 'acceleration' function 
a = 18t  18 