Section 12.11

Velocity and Acceleration

Describing the physics of motion along 3D curves.

Introduction

If is position, then:
- Velocity is the rate of change of position.
- Acceleration is the rate of change of velocity.
Acceleration can be split into two parts: one that changes speed (Tangential) and one that changes direction (Normal).

Interactive: Components of Acceleration

Velocity (v) is tangent to the path. Acceleration (a) points inward (toward the force).

Key Formulas

Basic Definitions

Acceleration Components

(Tangential)

(Normal)

Worked Examples

Example 1: Velocity from Position (Level 1)

Given , find and .

Velocity (First Derivative):

(Using product rule for z-component).

Acceleration (Second Derivative):
.

Example 2: Integration (Level 2)

Given acceleration , initial velocity , and initial position , find position .

Step 1: Find v(t)

Integrate :.
Use :.
.
.

Step 2: Find r(t)

Integrate :.
Use :.
.

Example 3: Acceleration Components (Level 3)

Find and for at .

, .
At : , .

Tangential: .
Normal: .
Cross: . Magnitude = 2.
.