Overview
This lecture analyzes 3D vector-valued functions through a spaceship trajectory, focusing on position, velocity, acceleration, and related calculations.
Trajectory Analysis
- The position vector is given by (cos t, sin t, e^(–t)) for t ≥ 0.
- X and Y components (cos t, sin t) trace a unit circle in the xy-plane.
- Z component (e^(–t)) models exponential decay, starting at z = 1 and approaching zero as t increases.
- The trajectory forms a downward spiral (helix) with a decreasing pitch as z decreases.
Velocity and Speed Calculations
- Velocity is the derivative of position: (–sin t, cos t, –e^(–t)).
- Speed is the magnitude of velocity: √[sin²t + cos²t + (e^(–t))²], which simplifies to √[1 + e^(–2t)].
- At t = 0, speed = √2.
Acceleration and its Magnitude
- Acceleration is the derivative of velocity: (–cos t, –sin t, e^(–t)).
- Magnitude of acceleration: √[cos²t + sin²t + (e^(–t))²] = √[1 + e^(–2t)].
- Maximum acceleration occurs at t = 0; magnitude decreases as t increases.
Piecewise Trajectory and Intersection with XY-Plane
- After t = 1, the trajectory switches to a straight line: l(t) = position at t=1 + velocity at t=1 × time.
- To find the intersection with the xy-plane (z = 0), set l(t)'s z-component to 0 and solve for time and coordinates.
Key Terms & Definitions
- Vector-valued function — A function with outputs as vectors, often parameterizing curves in space.
- Helix — A 3D spiral with a circular projection and varying height.
- Velocity — The derivative of position, a vector showing direction and rate of movement.
- Speed — The magnitude (scalar) of the velocity vector.
- Acceleration — The derivative of velocity, indicating change in motion.
Action Items / Next Steps
- Practice finding the intersection point of the straight-line trajectory with the xy-plane.
- Review how to compute velocity and acceleration from a vector-valued position function.