Motion Review 1-D Formulas: time t ≥ 0 Position Function: r(t) Tells you the location of a moving object

Velocity Function: v(t) = drdt

Tells you how fast the object is moving AND Tells you the direction in which the object is moving

Speed Function: v(t) Tells you only how fast the object is moving

Acceleration Function: a(t) =dvdt

= d2rdt 2

Tells you how fast the velocity is changing AND Tells you if the velocity is increasing or decreasing Example: Position Function: r(t) = t 2 − 4t − 3 Velocity Function: v(t) = Speed Function: v(t) = Acceleration Function: a(t) = When t = 1:

2-D Formulas: time t ≥ 0 Position Function:

r (t) = x(t), y(t) Tells you the location of a moving object Velocity Function:

v(t) Tells you how fast the object is moving AND

Tells you the direction in which the object is moving Speed Function:

v(t) 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 Plot: f (3) = Vector-Valued Functions:

r (t) = x(t), y(t) Domain (input) Range (output) Example:

r (t) = t 2 ,2t Plot:

r (3) =

Calculus Derivatives Review Real-Valued Functions:

Definition: ′f (t) = df

dt= limt→0

ft

= limt→0

f (t +t)− f (t)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:

Definition:

′r (t) = drdt

= limt→0

rt

= limt→0

r (t +t)− r (t)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: f (t)dt = F(t)+C where ′F (t) = f (t)∫

Definite Integrals: f (t)dta

b

∫ = F(b)− F(a) How do you find it? Use the antiderivative rules and the Fundamental Theorem of Calculus Example:

f (t) = t 2

t 2dt =∫

t 2dt1

2

∫ =

Vector-Valued Functions: Indefinite Integrals:

r (t)dt∫ = x(t), y(t) dt∫ = x(t)dt, y(t)dt∫∫ Definite Integrals:

r (t)dta

b

∫ = x(t), y(t) dta

b

∫ = x(t)dta

b

∫ , y(t)dta

b

∫

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∫ =

cos4t,sin 4t dt0

π4∫

Motion in 2-D Formulas: time t ≥ 0 Position Function:

r (t) = x(t), y(t) 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:

v(t) = drdt

=

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?

Speed Function:

v(t) = Tells you only how fast the object is moving Length of the velocity vector (speed is a scalar)

Acceleration Function:

a(t) = dvdt

= d2rdt 2

=

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 At t = 1 At t = 2 Position

r (0) = r (1) =

r (2) = Velocity

v(0) = v(1) =

v(2) = Acceleration

a(0) = a(1) =

a(2) = Speed

v(0) =

v(1) =

v(2) =

Motion in 3-D Formulas: time t ≥ 0 Position Function:

r (t) = x(t), y(t), z(t) 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:

v(t) = drdt

=

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 Speed Function:

v(t) = Tells you only how fast the object is moving Length of the velocity vector (speed is a scalar)

Acceleration Function:

a(t) = dvdt

= d2rdt 2

=

Tail at (x(t), y(t), z(t))

Example:

r (t) = x(t), y(t), z(t) = 2sin t,2 cos t,2t Position function:

r (t) = Velocity function:

v(t) = Acceleration function:

a(t) = Speed function:

v(t) = What is the path? At t = 0 At t = 1 At t = 2

r (0) =

r (1) ≈ 1.7,1.1,2 r (2) ≈ 1.8,−.8, 4

v(0) =

v(1) ≈ 1.1,−1.7,2 v(2) ≈ −.8,−1.8,2

a(0) =

a(1) ≈ −1.7,−1.1,0 a(2) ≈ −1.8,.8,0

v(0) =

v(1) ≈ 2.8 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) = Speed function:

v(t) = What is the path? At t = 0 At t = 1 At t = 2

r (0) =

r (1) ≈ .9,−.4 r (2) ≈ −.8,−.7

v(0) =

v(1) ≈ −.8,−1.8 v(2) ≈ −1.3,1.5

a(0) =

a(1) ≈ −3.6,1.7 a(2) ≈ 3.0,2.6

v(0) =

v(1) = 2 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) = Speed function:

v(t) = What is the path? At t = 0 At t = 1 At t = 2

r (0) =

r (1) ≈ .8,.5 r (2) ≈ −.8,−.7

v(0) =

v(1) ≈ 1.1,−1.7 v(2) ≈ −2.6,3.0

a(0) =

a(1) ≈ −2.3,−3.8 a(2) ≈ 10.8,12.0

v(0) =

v(1) = 2 v(2) = 4

Example:

r (t) = x(t), y(t) = 2t − 2sin t,2 − 2cos t Position function:

r (t) = Velocity function:

v(t) = Acceleration function:

a(t) = Speed function:

v(t) = What is the path? At t = 0 At t = 1 At t = 2

r (0) =

r (1) ≈ .3,.9 r (2) ≈ 2.2,2.8

v(0) =

v(1) ≈ .9,1.7 v(2) ≈ 2.8,1.8

a(0) =

a(1) ≈ 1.7,1.1 a(2) ≈ 1.8,−.8

v(0) =

v(1) ≈1.9 v(2) ≈ 3.4