SECT. 3-A
POSITION, VELOCITY,
AND ACCELERATION
Position function - x(t)
gives the location of an object at time t, usually s(t), x(t) or y(t)
Velocity - v(t)  x (t)

The rate of change( derivative) of position, usually v(t)
Acceleration - a(t)  v (t)  x (t)
The rate of change ( derivative) of velocity usually a(t)

Average velocity 
distance
f (b)  f (a)

time
ba
1) If s(t )  t  3t  5t
find v(t) and a(t).
3
2

2) The position of a particle moving along the x - axis at
time t is given by
x(t)  t 3 11t 2  24t . Find the particles velocity and
acceleration at t = 5.
Velocity
Acceleration
Speed – The absolute value of
velocity otherwise known as the
magnitude of velocity
speed  x' (t )  v(t )
Velocity
Positive - the particle is moving to the right
Negative - the particle is moving to the left
Zero - the particle has momentarily stopped
or is changing direction ( must have a sign
change)
3) Find where the object changes direction if
x(t)  t 3 12t 2  36t 18
Find where v(t) = 0

interval
time
V(t)
Result
Acceleration and Velocity
• If the sign of acceleration is the same as velocity, the
speed of the particle is ______________ (the two are
working together)
• If the sign of the acceleration is opposite that of velocity,
the speed of the particle is _________________ (the two
are working against each other)
4) Given the same position function as #3
x(t )  t 3 12t 2  36t  18
find the interval during which the
particle is slowing down.
Interval
t
velocity
acceleration
result
3
2
x
(
t
)

2
t

21
t
 60t  3
5) Given
find the
interval during which the particle is
speeding up.
Interval
t
velocity
acceleration
result
Distance vs. Displacement
• Displacement- change in position ( final position
minus original position)
• Distance- the total distance travelled by an object
in the time interval even if duplicated
6) Find the DISTANCE traveled by the particle
whose position is given by x(t )  t 4  8t 2
on the interval (0,4).
Distance NOT displacement!!
v(t )  4t 3 16t  0
4t (t 2  4)  0
t  0,2,2
Consider two
intervals
D  x(2)  x(0)  x(4)  x(2)
 (16)  0  128 (16)
 160
7) If x(t )  t  6t
a) find the DISTANCE traveled by the particle
on the interval (2,4).
2
b) Find the DISPLACEMENT on the interval
(1,5)
8) The graph shows the position
function of a radio controlled
car
a) Was the car going faster at
B or at C?
b) When was the car stopped?
c) At which point was the car’s velocity
the greatest?
d) At which point was the car’s speed
decreasing?
Homework
PVA Worksheet and
worksheet 3-A