Calculus notes – Rectilinear Motion
Name: _______________________
Rectilinear motion – object moving in either direction along a coordinate line (x-axis, y-axis, or an axis
that is inclined at some angle)
Position – s(t)
Velocity – the rate at which the
coordinate of a particle changes
with time and in what direction
v(t) = s’(t) = ds/dt
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When s(t) > 0, the particle is
on the positive side of the saxis
When s(t) < 0, the particle is
on the negative side of the saxis
When s(t) is increasing, the
particle moves in the positive
direction
When s(t) is decreasing, the
particle moves in the
negative direction.
When s(t) is concave up, the
acceleration is positive.
When s(t) is concave down,
the acceleration is negative
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When v(t) > 0, s(t) is
increasing, the particle
moves in the positive
direction.
When v(t) < 0, s(t) is
decreasing, the particle
moves in the negative
direction.
When v(t) = 0, the particle is
momentarily stopped.
Acceleration – the rate at which
the velocity of a particle changes
with time
a(t) = v’(t) = dv/dt
= s”(t) = d2s/dt2
 When a(t) > 0, the velocity is
increasing.
 When a(t) < 0, the velocity is
decreasing.
Speed = |velocity|
Speed describes how fast an
object is moving without
regard to direction.
A particle speeds up when its
velocity and acceleration
have the same sign.
A particle is slowing down
when they have opposite
signs.
Example: s(t) = t3 – 6t2, 0 < t < 8
Position versus time curve
s(t)
s(t) = t3 – 6t2
Velocity versus time curve
v(t)
v(t) =
Describe the motion of the particle
Acceleration versus time curve
a(t)
a(t) =
Verify with calculator in parametric mode
x1T = T3 – 6T2
y1T = 5 or T
0<T<8
-40 < x < 60
0<y<8
Example: s(t) = 2t3 – 21t2 + 60t + 3, 0 < t < 8
Describe the motion of the particle without a calculator.
Describe the motion of the particle using a calculator and the function s(t)
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7/2
Verify in parametric mode
x1T = 2T3 – 21T2 + 60T + 3
y1T = 5 or T
0<T<8
0 < x < 75
0<y<8
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