Displacement, Velocity, Acceleration, and Second Derivatives

Displacement, Velocity,
Acceleration, and Second
Derivatives

Displacement
● Displacement is the change of distance from its original position.
● Displacement is a vector quantity, as it has a direction.
From this lesson onwards, let displacement represented by D, where D = d(t). The
function has the variable of time, t as a parameter.
D = d(t)
When d(t) = 0, the object is at its original position.

Velocity
● Velocity is the rate of change of displacement (with respect to time). It
measures how much displacement is made over a period of time.
● Velocity tells information about the movement of the object.
V = v(t) = d’(t)
When v(t) = 0, the object is not moving.
When v(t) > 0, the object is moving away from the origin.
When v(t) < 0, the object is moving towards the origin.

Acceleration
● Acceleration is the rate of change of velocity (with respect to time). It
measures how much velocity is increased or decreased over a period of time.
A = a(t) = v’(t) = d’’(t)
When a(t) = 0, the object is moving with constant
velocity.
When a(t) > 0, the object undergoes increasing velocity.
When a(t) < 0, the object undergoes decreasing velocity.

Relationship
Since the three functions relate to each
other, they can be graphed with a
single displacement equation.
Let d(t) = t2 - 4
v(t) = d’(t) = 2t
a(t) = v’(t) = d’’(t) = 2
D = d(t)
V = v(t)
A = a(t)

Analysis
Displacement
● Object is at its initial position at t =
-2. and t = 2.
● The answer t = -2 is rejected since
time cannot be negative.
Velocity
● Object is moving away from the
origin.
Acceleration
● Object is accelerating at a 2 m/s2 at
all time.
D = d(t)
V = v(t)
A = a(t)

Analysis
Velocity
The object changed its movement
(from moving away to moving towards
the origin) at t = 3.
Object is moving towards origin at t < 3,
and moving away from origin at t > 3.
Acceleration
Acceleration is constant at all times.
Object is accelerating towards the
origin at t < 2, and accelerating away at
t > 2.
V = v(t)
D = d(t)

Displacement, Velocity, Acceleration, and Second Derivatives

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