3. Acceleration
a(t) = v'(t) =
dv
dt
OR
a(t) = s"(t) =
d2
s
dt2
Speeding up and Slowing down
v(t) > 0 and a(t) > 0
v(t) < 0 and a(t) < 0
Particle is speeding up
v(t) > 0 and a(t) < 0
v(t) < 0 and a(t) > 0
Particle is slowing down
4. Let’s see how to
calculate this !!
Let be the position function of a particle moving along an s-axis, where is
is in meters and t is in seconds. Find the velocity, speed and acceleration functions, and
show the graphs of position, velocity, speed and acceleration versus time.
s(t)= t3
-6t2
v(t) = 3t2
-12t v(t) = 3t2
-12t
s(t)= t3
-6t2
a(t)= 6t -12
5. Analyzing position versus time curve
Position versus Time
Curve
Characteristics of
the curve at t = to
Behavior of the Particle at t = to
• s(to) > 0
• Positive slope
• Concave down
• Particle is a the positive side of the origin
• Particle is moving in the positive dir.
• Velocity is decreasing
• Particle is slowing down
• s(to) > 0
• Negative slope
• Concave down
• Particle is a the positive side of the origin
• Particle is moving in the negative dir.
• Velocity is decreasing
• Particle is speeding up
• s(to) < 0
• Negative slope
• Concave up
• Particle is a the negative side of the origin
• Particle is moving in the negative dir.
• Velocity is increasing
• Particle is slowing down
• s(to) > 0
• Zero slope
• Concave down
• Particle is a the positive side of the origin
• Particle is momentarily stopped
• Velocity is decreasing
to
to
to
to
6. Practice Time !!!
Suppose that the position function of a particle moving
on a coordinate line is given by
.
Analyze the motion of the particle for t > 0. Summarize
the information schematically.
s(t)= 2t3
-21t2
+60t +3
v(t) = 6t2
-42t +60 = 6 t -2
( ) t -5
( )
a(t) =12t - 42 =12 t -
7
2
æ
è
ç
ö
ø
÷
v(t)
a(t)
· · ·
·
·
·
·
·
0 2 7/2 5
0 2 7/2 5
+ + + + + + + + + 0 - - - - - - - - - - - - - - - - - - - -0 + + + + + + +
- - - - - -- - - - - - - - - - - - - - - - - -0 + + + + + + + + + + + + + + +
Slowing down Speeding up Slowing down Speeding up
7. Not Done Yet !!!
s(0) = 3
s(2)= 55
s
7
2
æ
è
ç
ö
ø
÷ = 41.5
s(5) = 28
· · · · ·
0 3 28 41.5 55 s(t)
·
·
·
·
t = 0 t = 2
t = 7/2
t = 5
Slowing down
Speeding up
Slowing down
Speeding up
