# Calculus for Game Motion

Use slope, velocity, acceleration, area, and frame time to reason about game motion before tuning jumps, slides, camera movement, projectiles, or drone-style motion values.

## Outcome
Connect position, velocity, acceleration, and frame time so a motion change can be predicted before it is tuned.

## Safe first step
Write the axis, units, starting position, velocity, acceleration, and time step before changing a physics value.

## Ladder steps
### 1. Name the axis
Choose the horizontal, vertical, camera, projectile, or sensor axis before changing a motion value.

Check: Your note says which direction is positive and what unit the position uses.

### 2. Separate position from velocity
Position is where the object is; velocity is how that position changes through time.

Check: You can explain whether a bug is about location, speed, or direction.

### 3. Let acceleration change velocity
Gravity, thrust, boost, braking, and drag-style effects usually change velocity first.

Check: Your note predicts how velocity changes before checking position.

### 4. Read the graph before tuning
Slope and area explain why motion curves bend, flatten, or reverse.

Check: You can sketch the expected curve before changing the game value.

## Examples
### Read slope as speed
```sh
position changes -> velocity clue
```
Expected signal: A steeper position curve means larger velocity

### Read acceleration as velocity change
```sh
velocity changes -> acceleration clue
```
Expected signal: Constant negative acceleration bends vertical motion downward

### Check one update step
```sh
frame time matters
```
Expected signal: A smaller time step should make each single-frame change smaller

## Common traps
- Tuning jump height without writing units or frame time.
- Confusing speed with signed velocity.
- Changing position directly when velocity or acceleration should own the motion.
- Forgetting that some game coordinate systems put down as positive y.
- Treating a game simulation value as proof of real-world safety.

## Practice task
Use the widget with one jump or slide example. Change acceleration once, predict the graph, then copy the note and explain whether velocity or position changed first.

## Next steps
- Save the Obsidian note with [[Position]], [[Velocity]], [[Acceleration]], [[Frame Time]], [[Slope]], [[Area Under Curve]], [[Projectile Motion]], [[Game Physics]], and [[Godot]] backlinks.
- Use vectors when the motion direction is unclear.
- Use trigonometry when the motion starts from an angle.
- Use interpolation when the motion needs a smooth blend between states.
- Use sensor statistics when recorded motion samples are noisy.

## Related
- [Maker Math routes](/topics/math/)
- [Vectors for maker projects](/learn/vectors-for-maker-projects/)
- [Trigonometry for rotation and layout](/learn/trigonometry-for-rotation-and-layout/)
- [Interpolation for motion and controls](/learn/interpolation-for-motion-and-controls/)
- [Build a first Godot game loop](/projects/build-first-godot-game-loop/)
- [Basic statistics for sensor logs](/learn/data/basic-statistics-for-sensor-logs/)
- [Turn a source video into notes](/video-notes/)
- [Review Anki-compatible decks](/decks/)

## Obsidian backlinks

Use these wiki links to connect this note inside a local maker vault:

- [[TopicLadder]]
- [[Maker Learning]]
- [[Calculus for Game Motion]]
- [[Maker Math]]
- [[math]]
- [[applied-foundation]]
- [[Name the axis]]
- [[Separate position from velocity]]
- [[Let acceleration change velocity]]
- [[Read the graph before tuning]]
- [[Maker Math routes]]
- [[Vectors for maker projects]]

## Source and next routes

Source: https://topicladder.com/learn/math/calculus-for-game-motion/

- [Maker Math routes](/topics/math/)
- [Vectors for maker projects](/learn/vectors-for-maker-projects/)
- [Trigonometry for rotation and layout](/learn/trigonometry-for-rotation-and-layout/)
- [Interpolation for motion and controls](/learn/interpolation-for-motion-and-controls/)
- [Build a first Godot game loop](/projects/build-first-godot-game-loop/)
- [Basic statistics for sensor logs](/learn/data/basic-statistics-for-sensor-logs/)
- [Turn a source video into notes](/video-notes/)
- [Review Anki-compatible decks](/decks/)