3. Acceleration
dv
a(t) = v'(t) =
dt
OR
d 2s
a(t) = s"(t) = 2
dt
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 s(t) = t 3 - 6t 2 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) = t 3 - 6t 2
v(t) = 3t 2 -12t
v(t) = 3t 2 -12t
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 s(t) = 2t 3 - 21t 2 + 60t + 3
.
Analyze the motion of the particle for t > 0. Summarize
the information schematically.
v(t) = 6t 2 - 42t + 60 = 6 (t - 2) (t - 5)
æ 7ö
a(t) =12t - 42 =12 ç t - ÷
è 2ø
0
·
2
7/2
·
·
2
7/2
5
·
v(t)
+ + + + + + + + + 0 - - - - - - - - - - - - - - - - - - - -0 + + + + + + +
0
·
5
·
·
·
- - - - - -- - - - - - - - - - - - - - - - - -0 + + + + + + + + + + + + + + + a(t)
Slowing down
Speeding up Slowing down Speeding up
7. Not Done Yet !!!
s(0) = 3
æ 7ö
s ç ÷ = 41.5
è2ø
s(2) = 55
s(5) = 28
Speeding up
t=5 ·
t=0
·
· ·
0
3
t = 7/2
·
Slowing down Speeding up t = 2
·
Slowing down
·
28
·
41.5
·
55
s(t)
