One-Dimensional Motion

Core Sentence / 核心句

In one-dimensional motion, velocity is the derivative of position, acceleration is the derivative of velocity, and speed is the magnitude of velocity.

中文理解：一维运动中，位置求导得到带方向的速度，速度再求导得到加速度；速率只保留速度大小。

Formula / 公式

$$v(t)=x^{\prime }(t),a(t)=v^{\prime }(t)=x^{\prime \prime }(t),\text{speed}=|v(t)|.$$

Sign Map / 符号判断

• $v>0$: moving right
• $v<0$: moving left
• $a>0$: velocity increasing
• $a<0$: velocity decreasing
• $va>0$: speed increasing
• $va<0$: speed decreasing

Watch Out / 易错点

• Velocity decreasing is not the same as speed decreasing.
• A particle may have $v=0$ while $a\ne 0$.
• On a velocity graph, height gives direction and slope gives acceleration.
• At $v=0$, inspect nearby behavior before applying the sign shortcut.

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第4次正课 · Four Motion Quantities
  • position, velocity, acceleration, and speed form one connected derivative system

Connected Notes / 连接

• 微积分BC第4次正课-反函数高阶导数与运动应用
• Derivatives
• Instantaneous Rate of Change
• Higher-Order Derivatives
• Second Derivative
• Acceleration