Space Curve Slides Space Curves
Defining Vector-Valued Functions
MISSION: KINEMATICS_01
The Helix Mystery
Visual Inquiry
Imagine a particle moving in a spiral. It climbs vertically as it circles the center.
Think-Pair-Share:
If you looked at its shadow on the floor (xy-plane), what shape would you see?
What about its shadow on the wall (yz-plane)?
[3D HELIX VISUALIZATION]
Defining Vector Functions
A vector-valued function maps a real number parameter \( t \) to a vector in space.
\[ \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \]
Components
\( f, g, h \) are scalar-valued component functions.
Domain
The intersection of the domains of all component functions.
Range
A set of vectors that trace out a space curve.
Finding the Domain
Example Problem
\[ \mathbf{r}(t) = \left\langle \ln(t+1), \frac{1}{t-2}, \sqrt{5-t} \right\rangle \]
1 Identify constraints for each component.
2 Find the intersection of all intervals.
Step-by-Step Analysis:
\( \ln(t+1) \) \( t > -1 \)
\( \frac{1}{t-2} \) \( t \neq 2 \)
\( \sqrt{5-t} \) \( t \leq 5 \)
Final Domain: \( (-1, 2) \cup (2, 5] \)
Limits and Continuity
The limit of a vector-valued function is calculated component-wise:
\[ \lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle \]
Continuity
\( \mathbf{r}(t) \) is continuous at \( a \) if each component function is continuous at \( a \).
Warning
The limit only exists if the limits for all components exist individually.
Next Steps: Vector Mapping
You are now ready to map space curves. We will transition from simple lines to complex spirals and intersections.
Skill Check
Identify domains of trig-based vector functions.
Visual Goal
Predict the 3D shape from 2D component projections.
Vector Mapping Worksheet Vector Mapping
MISSION: KINEMATICS_01 / SPACE CURVES
NAME:
DATE:
Objective
In this exercise, you will define vector-valued functions, analyze their domains, and predict their geometric representations by projecting components onto coordinate planes.
1
Domain Identification
Find the domain of the following vector-valued functions. Show your interval constraints for each component.
1.1) \(\mathbf{r}(t) = \langle \sqrt{t+4}, \ln(10-t), t^2 \rangle\)
Component Constraints:
Final Domain (Interval Notation):
1.2) \(\mathbf{r}(t) = \left\langle \sin(t), \frac{1}{\sqrt{t-1}}, e^{2t} \right\rangle\)
Component Constraints:
Final Domain (Interval Notation):
2
Parametric Translation
Consider the vector function \(\mathbf{r}(t) = \langle 3\cos(t), 3\sin(t), 2t \rangle\). This represents a circular helix.
A) Write the parametric equations for this curve.
B) Eliminate the parameter \(t\) for the \(x\) and \(y\) components. What 2D shape is formed in the \(xy\)-plane?
C) Sketch the projection on the \(xy\)-plane.
3
Limiting Behavior
Evaluate the following limit or explain why it does not exist.
\[ \lim_{t \to 0} \left\langle \frac{\sin(t)}{t}, \frac{e^t - 1}{t}, \cos(t) \right\rangle \]
Show your work for each component limit below:
Tangent Vector Slides Tangent Vectors
Differentiation & Velocity
MISSION: KINEMATICS_02
Instantaneous Heading
Kinematic Inquiry
Flight Path \(\mathbf{r}(t)\)
A drone follows a 3D path. At any second, we know its position \(\mathbf{r}(t)\).
The Challenge:
How do we calculate the direction it is pointing?
How do we calculate its speed?
The Derivative Vector
If \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), then its derivative is:
\[ \mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle \]
Geometry
\(\mathbf{r}'(t)\) is a tangent vector to the curve at point \(P\). It points in the direction of increasing \(t\).
Physics
If \(\mathbf{r}(t)\) is position, then \(\mathbf{r}'(t) = \mathbf{v}(t)\) is the velocity vector.
The Unit Tangent Vector
Often we only care about the direction of travel, not the speed. We normalize the tangent vector:
\[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|} \]
1
Differentiate each component.
2
Calculate the magnitude (speed).
3
Divide vector by magnitude.
Equation of the Tangent Line
The tangent line at \(t = t_0\) passes through point \(\mathbf{r}(t_0)\) in direction \(\mathbf{r}'(t_0)\).
Vector Equation
\[ \mathbf{L}(s) = \mathbf{r}(t_0) + s \mathbf{r}'(t_0) \]
Key Parameters:
Position: \(\mathbf{r}(t_0)\)
Direction: \(\mathbf{r}'(t_0)\)
Parameter: \(s\) (distinct from \(t\))
Differentiation Rules
Dot Product Rule
\[ \frac{d}{dt} [\mathbf{u} \cdot \mathbf{v}] = \mathbf{u}' \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{v}' \]
Cross Product Rule
\[ \frac{d}{dt} [\mathbf{u} \times \mathbf{v}] = \mathbf{u}' \times \mathbf{v} + \mathbf{u} \times \mathbf{v}' \]
Crucial Note
The order of terms in the Cross Product Rule must be maintained because the cross product is not commutative!
Velocity Vector Lab Worksheet Velocity Vector Lab
MISSION: KINEMATICS_02 / DIFFERENTIATION
NAME:
DATE:
1
Derivative Calculations
Find the derivative vector \(\mathbf{r}'(t)\) for each position function.
1.1) \(\mathbf{r}(t) = \langle e^{3t}, t^4, \sin(2t) \rangle\)
1.2) \(\mathbf{r}(t) = \langle t \ln(t), \sqrt{1+t^2}, \arctan(t) \rangle\)
2
The Tangent Line
Consider the curve \(\mathbf{r}(t) = \langle t, t^2, t^3 \rangle\) (the twisted cubic).
A) Find the position vector \(\mathbf{r}(1)\) and the tangent vector \(\mathbf{r}'(1)\).
\(\mathbf{r}(1) =\)
\(\mathbf{r}'(1) =\)
B) Write the vector equation for the tangent line to the curve at the point (1, 1, 1).
Vector Equation \(\mathbf{L}(s) =\)
3
Speed & Unit Direction
A particle moves with position \(\mathbf{r}(t) = \langle 4\cos(t), 4\sin(t), 3t \rangle\). Find the following at \(t = \pi\):
A) The velocity vector \(\mathbf{v}(\pi)\):
B) The speed of the particle at \(t = \pi\):
C) The unit tangent vector \(\mathbf{T}(\pi)\):
Theory Extension
Suppose \(|\mathbf{r}(t)| = c\) (constant magnitude). Prove that \(\mathbf{r}(t) \cdot \mathbf{r}'(t) = 0\). Hint: Start by differentiating the expression \(\mathbf{r}(t) \cdot \mathbf{r}(t) = c^2\).
Proof Area:
Projectile Path Slides Module 03
Integration &
Projectile Motion
Reversing differentiation to model trajectories in space.
BALLISTICS_INIT_03
The Ballistics Challenge
Problem Context
A projectile hit a target 100 meters away. We know the launch angle and the impact location.
The Mission:
Work backward from acceleration (gravity) to determine the exact velocity vector at the moment of launch.
IMPACT
Vector Integration
Integration of a vector function is performed component-wise:
\[ \int \mathbf{r}(t) dt = \left\langle \int f(t) dt, \int g(t) dt, \int h(t) dt \right\rangle + \mathbf{C} \]
The Constant Vector \(\mathbf{C}\)
The constant of integration is a vector: \(\mathbf{C} = \langle C_1, C_2, C_3 \rangle\).
Definite Integrals
FTC applies: \(\int_a^b \mathbf{r}(t) dt = \mathbf{R}(b) - \mathbf{R}(a)\), where \(\mathbf{R}\) is the antiderivative.
Initial Value Problems (IVPs)
To find the specific path, we need Initial Conditions.
Velocity at \(t=0\)
\(\mathbf{v}(0) = \mathbf{v}_0\)
Position at \(t=0\)
\(\mathbf{r}(0) = \mathbf{r}_0\)
The Integration Chain
Acceleration
\(\mathbf{a}(t)\)
Velocity
\(\int \mathbf{a}(t) dt = \mathbf{v}(t)\)
Position
\(\int \mathbf{v}(t) dt = \mathbf{r}(t)\)
Deriving Projectile Motion
Starting from gravity \(\mathbf{a}(t) = \langle 0, 0, -g \rangle\):
1. Velocity Vector
\[ \mathbf{v}(t) = \langle v_0 \cos \theta, v_0 \sin \theta, -gt + v_{0z} \rangle \]
2. Position Vector
\[ \mathbf{r}(t) = \langle (v_0 \cos \theta)t, (v_0 \sin \theta)t, -\frac{1}{2}gt^2 + v_{0z}t + z_0 \rangle \]
Notice how horizontal velocity remains constant while vertical velocity is linear.
Ready for Forensics?
We will now use these vector integration techniques to solve real-world trajectory problems, starting with our forensic ballistics case.
Integration
Component by component.
Constants
Determined by initial state.
Trajectory
The path of the particle.
Ballistics Forensics Worksheet Ballistics Forensics
MISSION: KINEMATICS_03 / INTEGRATION
NAME:
DATE:
Case File: Projectile Trajectory
A forensic team has recovered data from a high-speed projectile launch. Your task is to reconstruct the position function from the acceleration data and initial conditions to find the point of origin.
1
Integrating the Path
A projectile is launched with acceleration \(\mathbf{a}(t) = \langle 0, 0, -10 \rangle\) \(m/s^2\).
At \(t = 0\), the initial velocity is \(\mathbf{v}(0) = \langle 20, 5, 30 \rangle\) \(m/s\).
A) Find the velocity vector \(\mathbf{v}(t)\). Show the integration of each component.
Calculations Area:
B) Find the position vector \(\mathbf{r}(t)\). Show the integration of your velocity function.
Calculations Area:
2
Mission Analysis
Using your position function \(\mathbf{r}(t)\) from Part 1, answer the following questions:
2.1) At what time \(t\) does the projectile hit the ground (where \(z = 0\))?
2.2) What is the coordinates \((x, y)\) of the impact point?
2.3) What is the speed of the projectile at the moment of impact?
Displacement Evaluation
Calculate the total displacement vector from \(t=0\) to \(t=2\) for a different object moving with velocity:
\[ \int_0^2 \langle 3t^2, 2t, \pi \cos(\pi t) \rangle dt \]
Work & Final Vector:
Terminal Velocity Protocols Active
UNIT_03_BALLISTICS_REPORT
Curve Length Slides Curve Length
Arc Length & Parametrization
MISSION: GEOMETRY_04
The Path Paradox
Thought Experiment
Particle A moves at a constant 10 m/s.
The Question:
If they both trace the exact same spiral from point \(P\) to point \(Q\), does the length of the path depend on how fast they move?
Two Speeds, One Geometry
The Arc Length Integral
The length of a curve from \(t=a\) to \(t=b\) is the integral of speed:
\[ L = \int_a^b |\mathbf{r}'(t)| dt \]
In 3D Components:
\[ \int_a^b \sqrt{(f')^2 + (g')^2 + (h')^2} dt \]
This formula effectively "unrolls" the curve into a straight line and measures it.
The Arc Length Function \(s(t)\)
To treat distance as the independent variable, we define the distance traveled up to time \(t\):
\[ s(t) = \int_{t_0}^t |\mathbf{r}'(u)| du \]
Fundamental Property:
Goal:
Reparametrization by Arc Length
When we write \(\mathbf{r}(s)\) instead of \(\mathbf{r}(t)\), we are describing the pure geometry of the curve.
The Magic Rule
For any curve parametrized by arc length \(s\):
\[ |\mathbf{r}'(s)| = 1 \]
Physical Interpretation
The particle moves at unit speed.
Distance along the path equals the parameter change.
Essential for defining Curvature.
The Geometry Blueprint
By integrating speed and reparametrizing, we separate "how we move" from "where we go." This is the key to advanced curve analysis.
Practice Goal 1
Calculate the exact length of helix and power-function segments.
Practice Goal 2
Convert standard time parameters into arc length units.
Arc Length Explorer Worksheet Arc Length Explorer
MISSION: GEOMETRY_04 / PARAMETRIZATION
NAME:
DATE:
1
The Distance Integral
Calculate the exact length of the curve over the given interval. Show the magnitude calculation and the integration steps.
1.1) \(\mathbf{r}(t) = \langle 3t, \sin(4t), \cos(4t) \rangle\) for \(0 \leq t \leq \pi\)
Step 1: Calculate \(|\mathbf{r}'(t)|\)
Step 2: Evaluate the Integral
1.2) \(\mathbf{r}(t) = \langle t^2, \frac{2}{3}t^3, 1 \rangle\) for \(0 \leq t \leq 2\)
Calculations Area:
2
Mapping Distance to Time
Find the arc length function \(s(t)\) for the curve starting from the given initial point \(t_0\).
\[ \mathbf{r}(t) = \langle e^t \cos(t), e^t \sin(t), e^t \rangle \quad \text{starting from } t_0 = 0 \]
A) Calculate the speed \(|\mathbf{r}'(t)|\):
B) Integrate to find \(s(t) = \int_0^t |\mathbf{r}'(u)| du\):
3
Geometric Reparametrization
Use your result from Part 2 (B) to reparametrize the curve \(\mathbf{r}(t)\) with respect to arc length \(s\).
1. Solve for \(t\) in terms of \(s\):
2. Substitute \(t(s)\) into \(\mathbf{r}(t)\):
\(\mathbf{r}(s) = \)
Surveyor Grade Analysis
UNIT_04_GEOMETRY_PROTOCOLS
Acceleration Decomposition Slides Decomposing
Acceleration
Tangential & Normal Components
MISSION: DYNAMICS_05
The Racing Experience
Physical Inquiry
Imagine you are in a high-performance race car entering a sharp turn.
Observation:
You feel pushed back into your seat as the car speeds up.
You feel pushed sideways as the car whips around a corner.
Force Vector Analysis
Resolving Acceleration
Total acceleration \(\mathbf{a}\) can be broken into two orthogonal components:
\[ \mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N} \]
\(a_T\) : Tangential
Measures the change in speed.
\(a_T = \frac{d}{dt}|\mathbf{v}|\)
\(a_N\) : Normal
Measures the change in direction.
\(a_N = \kappa |\mathbf{v}|^2\)
Calculating the Components
Tangential (\(a_T\))
\[ a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|} \]
Proximity of acceleration to velocity vector.
Normal (\(a_N\))
\[ a_N = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|} \]
Amount of acceleration orthogonal to path.
Pythagorean Check:
\[ |\mathbf{a}|^2 = a_T^2 + a_N^2 \]
Visual Diagnostics
Straight Line
If the path is straight, \(a_N = 0\). Curvature is zero.
Uniform Speed
If speed is constant, \(a_T = 0\). Acceleration is purely centripetal.
Dynamic Motion
Both \(a_T\) and \(a_N\) are present. Speed and direction change.
The Final Turn
You have the tools to analyze any particle trajectory in the universe. Time to put these formulas to work on the track.
Task 1
Calculate Components
Task 2
Physical Insight
Acceleration Breakdown Worksheet Acceleration Breakdown
MISSION: DYNAMICS_05 / DECOMPOSITION
NAME:
DATE:
1
Scalar Components
Consider a particle moving along the curve \(\mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle\). Find the tangential and normal components of acceleration at any time \(t\).
A) Find \(\mathbf{v}(t)\) and \(\mathbf{a}(t)\):
B) Find the speed \(v = |\mathbf{v}(t)|\):
Tangential Component \(a_T\)
Formula: \(\frac{\mathbf{v} \cdot \mathbf{a}}{v}\)
Normal Component \(a_N\)
Formula: \(\frac{|\mathbf{v} \times \mathbf{a}|}{v}\)
2
Motion Analysis
2.1) Based on your calculations in Part 1, what can you conclude about the speed of the particle? Justify your answer using \(a_T\).
2.2) Curvature \(\kappa\) can be found using the formula \(\kappa = \frac{a_N}{|\mathbf{v}|^2}\). Use your results to calculate the curvature of the helix.
Trajectory Diagnostic
Match the physical scenario to the correct acceleration component behavior.
A car moves at a constant speed of 80 mph on a circular track.
\(a_T = 0\) \(a_N \neq 0\)
A rocket accelerates vertically upward in a straight line.
\(a_T \neq 0\) \(a_N = 0\)
A cyclist coasts downhill on a winding mountain road, speeding up as they turn.
\(a_T \neq 0\) \(a_N \neq 0\)
Dynamics Propulsion Protocol
UNIT_05_DYNAMICS_RECAP
