Velocity
Def. Velocity is the change in position of an object over a change in time.
v avg =
P2  P1
(Average velocity)
t2  t1
Note: The velocity of an object is directional.
-It will be positive if the change in position is linear and the object is
moving to the right on the x-axis or up on the y-axis.
-It will be negative if the change in position is linear and the object is
moving to the left on the x-axis or down on the y-axis.
------------------------------------------------------------------------------------------Examples:
1) Find the average velocity of a particle that went from
x = 2 to x = 20 in the time interval from t = 1 to t = 4 seconds.
2) Find the average velocity of an object moving from
y = 50 to y = 30 in the time interval from t = 3 to t = 7.
-------------------------------------------------------------------------------------------------------If a particles movement is defined by a function s(t), the average velocity in an
interval of time [t, t + h] would be given by using the slope formula:
Average velocity = v avg =
s (t  h)  s (t ) s (t  h)  s (t ) P2  P1
=
=
t ht
h
t2  t1
3) If a particle moves along the x-axis according to the formula s(t) = 2t  3 ,
find the average velocity of the particle in the time interval t = [1, 4] seconds.
2
Instantaneous Velocity
As the time interval “h” gets smaller and smaller about a particular time ( h  0) ,
the average velocity approaches the instantaneous velocity at a particular time, thus
requiring the use of the limit definition.
s (t  h)  s (t )
h 0
h
------------------------------------------------------------------------------------------Instantaneous velocity = v(t) = s’(t) = lim
Examples:
1) If s(t) = 2t  3 , find the instantaneous velocity at t = 3 seconds.
2
2) A person bungee jumps from a height of 1024 ft. and falls toward the ground
2
in a straight line. In t seconds the person drops a distance of 16t
feet.
What is the function to determine their position? _______________________
What is their instantaneous velocity at t = 5 seconds?
Acceleration
Def. Acceleration is the change in the velocity of an object over a change in time.
V2  V1
a avg =
t 2  t1 (Average Acceleration)
Instantaneous Acceleration
As the time interval “h” gets smaller and smaller about a particular time ( h  0) ,
the average acceleration approaches the instantaneous acceleration at a particular
time, thus requiring the use of the limit definition.
Instantaneous acceleration = a(t) = v’(t) = lim
h 0
v(t  h)  v(t )
h
-------------------------------------------------------------------------------------------------------Example:
s(t) is a function that determines the position of a falling object over t seconds.
s(t )  480  16t 2
0,4
t = 1, 2, 3
Find:
a)
b)
c)
d)
e)
f)
the average velocity in the given interval.
the instantaneous velocity formula.
the instantaneous velocity for the 3 times given.
the average acceleration in the given interval.
the instantaneous acceleration formula.
the instantaneous acceleration for the 3 times given.