Motion Review 1-D Formulas : time t ≥ 0 Position Function: r (t ) Tells you the location of a moving object Velocity Function: v(t ) = dr dt 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 ) = dv dt = d 2 r dt 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 − 4 t − 3 Velocity Function: v(t ) = Speed Function: v(t ) = Acceleration Function: a(t ) = When t = 1 :
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r t dr v t dt v t - Virginia Tech · 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
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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:
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?
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