Motion
Review 1-D
Formulas: time t ≥ 0
r(t)
Position Function:
Tells you the location of a moving object
Velocity Function:
dr
v(t) =
dt
Tells you how fast the object is moving AND
Tells you the direction in which the object is moving
v(t)
Speed Function:
Tells you only how fast the object is moving
dv d 2 r
= 2
Acceleration Function: a(t) =
dt dt
Tells you how fast the velocity is changing AND
Tells you if the velocity is increasing or decreasing
Example:
Position Function:
Velocity Function:
Speed Function:
Acceleration Function:
When t = 1 :
r(t) = t 2 − 4t − 3
v(t) =
v(t) =
a(t) =
2-D
Formulas: time t ≥ 0

r (t) = x(t), y(t)
Position Function:
Tells you the location of a moving object
v(t)
Velocity Function:
Tells you how fast the object is moving AND
Tells you the direction in which the object is moving

v(t)
Speed Function:
Tells you only how fast the object is moving

Acceleration Function: a(t)
Has both magnitude and direction
What do they tell you?
Example:

r (t) = x(t), y(t) = t 2 , 2t
Tells you the location of an object moving on the path
described by the parametric equations:
x(t) = t 2
y(t) = 2t
What is the path?
Vector-Valued Functions
Review Real(scalar)-Valued Functions: f (t)
Domain (input)
Range (output)
Example:
f (t) = t 2
f (3) =
Plot:

r
Vector-Valued Functions: (t) = x(t), y(t)
Domain (input)
Range (output)

2
r
(t)
=
t
, 2t
Example:

r (3) =
Plot:
Calculus
Derivatives
Review Real-Valued Functions:
df
f
f (t +t) − f (t)
= lim
= lim
Definition: f ′(t) =

t→0
dt
t t→0
t
What does it mean?
Instantaneous rate of change of output f (t) with respect to
input t .
How do you use it?
Find the slope of the line tangent to the curve at the point
(t, f (t)) .
How do you find it?
Use the derivative rules whenever possible
Example:
f (t) = t 2
f ′(t) =
f ′(3) =
Vector-Valued Functions:




dr
r
r (t +t) − r (t)

r
(t)
=
=
lim
=
lim
Definition: ′
dt t→0 t t→0
t
What does it mean?

Instantaneous rate of change of output r (t) with respect to
input t .
How do you use it?
How do you find it?


r (t) = x(t), y(t) ⇒ r ′(t) = x ′(t), y′(t)
Example:

r (t) = t 2 , 2t

r ′(t) =

r ′(3) =
Integrals
Review Real-Valued Functions:
Indefinite Integrals:
Definite Integrals:
∫
∫ f (t)dt = F(t) + C
b
a
where F ′(t) = f (t)
f (t)dt = F(b) − F(a)
How do you find it?
Use the antiderivative rules and the Fundamental Theorem
of Calculus
Example:
f (t) = t 2
2
t
∫ dt =
∫
2
1
t 2 dt =
Vector-Valued Functions:
Indefinite Integrals:

∫ r (t)dt =
∫
x(t), y(t) dt =
∫ x(t)dt, ∫ y(t)dt
Definite Integrals:
∫
b
a

r (t)dt =
∫
b
a
x(t), y(t) dt =
∫
b
a
b
x(t)dt , ∫ y(t)dt
a
How do you find it?
Use the antiderivative rules and the Fundamental Theorem
of Calculus on the components of the vector.
Examples:
∫
t 3 ,t dt =
∫
e2t ,sin t dt =
∫
π
0
4
cos 4t,sin 4t dt
Motion in 2-D
Formulas: time t ≥ 0

r
(t) = x(t), y(t)
Position Function:
Tells you the location of a moving object
Plot as a position vector
Tail at (0, 0)
Tip traces the curve with parametric equations:
x = x(t)
y = y(t)
Velocity Function:

dr

v(t) =
=
dt
Tells you how fast the object is moving AND
Tells you the direction in which the object is moving
Plot at the point corresponding to time t
Tail at (x(t), y(t))
Tip points?

v(t) =
Speed Function:
Tells you only how fast the object is moving
Length of the velocity vector (speed is a scalar)

2
dv d r

a(t)
=
= 2 =
Acceleration Function:
dt dt
Plot at the point corresponding to time t
Tail at (x(t), y(t))
Tip points?
Example:

r (t) = x(t), y(t) = t 2 , 2t
Position function:

r (t) =
Velocity function:

v(t) =
Acceleration function:

a(t) =
Speed function:

v(t) =
What is the path if t ≥ 0 ?
At
 t=0
r (0) =
Position

v(0) =
Velocity

a(0)
=
Acceleration

v(0) =
Speed
At
 t =1
r (1) =

v(1) =

a(1) =

v(1) =
At
 t=2
r (2) =

v(2) =

a(2) =

v(2) =
Motion in 3-D
Formulas: time t ≥ 0

r
(t) = x(t), y(t), z(t)
Position Function:
Tells you the location of a moving object
Plot as a position vector
Tail at (0, 0, 0)
Tip traces the curve with parametric equations:
x = x(t)
y = y(t)
z = z(t)
Velocity Function:

dr

v(t) =
=
dt
Tells you how fast the object is moving AND
Tells you the direction in which the object is moving
Plot at the point corresponding to time t
Tail at (x(t), y(t), z(t))
Tip points tangent to the curve in the direction of the
motion

v(t)
=
Speed Function:
Tells you only how fast the object is moving
Length of the velocity vector (speed is a scalar)

2
dv d r

a(t)
=
= 2 =
Acceleration Function:
dt dt
Tail at (x(t), y(t), z(t))
Example:

r (t) = x(t), y(t), z(t) = 2 sin t, 2 cost, 2t
Position function:

r (t) =
Velocity function:

v(t) =
Acceleration function:

a(t) =

v(t) =
Speed function:
What is the path?
At t = 0

r (0) =

v(0) =

a(0) =

v(0) =
At t = 1

r (1) ≈ 1.7,1.1, 2

v(1) ≈ 1.1, −1.7, 2

a(1) ≈ −1.7, −1.1, 0

v(1) ≈ 2.8
At t = 2

r (2) ≈ 1.8, −.8, 4

v(2) ≈ −.8, −1.8, 2

a(2) ≈ −1.8,.8, 0

v(2) ≈ 2.8
More 2-D:

r (t) = x(t), y(t) = sin(2t), cos(2t)
Position function:

r (t) =
Velocity function:

v(t) =
Acceleration function:

a(t) =

v(t) =
Speed function:
What is the path?
At t = 0

r (0) =

v(0) =

a(0) =

v(0) =
At t = 1

r (1) ≈ .9, −.4

v(1) ≈ −.8, −1.8

a(1) ≈ −3.6,1.7

v(1) = 2
At t = 2

r (2) ≈ −.8, −.7

v(2) ≈ −1.3,1.5

a(2) ≈ 3.0, 2.6

v(2) = 2
Example:

r (t) = x(t), y(t) = sin(t 2 ), cos(t 2 )
Position function:

r (t) =
Velocity function:

v(t) =
Acceleration function:

a(t) =

v(t) =
Speed function:
What is the path?
At t = 0

r (0) =

v(0) =

a(0) =

v(0) =
At t = 1

r (1) ≈ .8,.5

v(1) ≈ 1.1, −1.7

a(1) ≈ −2.3, −3.8

v(1) = 2
At t = 2

r (2) ≈ −.8, −.7

v(2) ≈ −2.6, 3.0

a(2) ≈ 10.8,12.0

v(2) = 4
Example:

r (t) = x(t), y(t) = 2t − 2 sin t, 2 − 2 cost
Position function:

r (t) =
Velocity function:

v(t) =
Acceleration function:

a(t) =

v(t) =
Speed function:
What is the path?
At t = 0

r (0) =

v(0) =

a(0) =

v(0) =
At t = 1

r (1) ≈ .3,.9

v(1) ≈ .9,1.7

a(1) ≈ 1.7,1.1

v(1) ≈ 1.9
At t = 2

r (2) ≈ 2.2, 2.8

v(2) ≈ 2.8,1.8

a(2) ≈ 1.8, −.8

v(2) ≈ 3.4