# Kinematics: The relationship between Distance, Speed & Acceleration

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 anti-differentiation Class Notes | Blank Notes 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)

#### Mouse over below Distance function S=

#   Speed function V=

#   Acceleration Function  A=

# ### Find the speed of a car at t = 4 seconds, given the distance (metres) is defined by d = t2 - 6t + 15

Steps:

d = t2 - 6t +15

1) Differentiate to find 'speed' function

v = 2t - 6

2) Substitute t = 4 into the 'speed' function

v = 2t - 6
v = 2x4 - 6
v = 8 - 6

3) Speed at t = 4 sec

v = 2 m/s

### Find the acceleration of a car at t = 2 seconds, given the distance (metres) is defined by d = t2 - 6t + 15

Steps:

d = t2 - 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 = 3t3 - 9t2 + 18

Steps:

d = 3t3 - 9t2 + 18

1) Differentiate to find 'speed' function

v = 9t2 - 18t

2) Differentiate again to find the 'acceleration' function

a = 18t - 18

3) Substitute t = 2 into the 'acceleration' function

a = 18t - 18
a = 18x2 - 18
a = 18 m/s/s