📚 Deriving the Equations of Motion | 运动学方程的推导
In CIE A-Level Physics, the equations of motion describe uniformly accelerated motion in a straight line. They link initial velocity u, final velocity v, acceleration a, time t and displacement s. Understanding how they are derived helps you choose the right equation and avoid sign errors.
在 CIE A-Level 物理中，运动学方程描述匀加速直线运动。它们联系初速度 u、末速度 v、加速度 a、时间 t 和位移 s。理解它们的推导过程有助于你选择正确的方程并避免符号错误。
1. Key Quantities and Conventions | 关键物理量与符号约定
Before deriving the equations, define the five quantities clearly. The symbol s is displacement, a vector quantity measured in metres. The symbols u and v are initial and final velocities, measured in m s⁻¹. Acceleration a is measured in m s⁻², and time t is measured in s.
在推导方程之前，先清晰定义五个物理量。符号 s 表示位移，是矢量，单位为米。符号 u 和 v 分别表示初速度和末速度，单位为 m s⁻¹。加速度 a 的单位为 m s⁻²，时间 t 的单位为 s。
In one-dimensional problems, choose a positive direction and apply it consistently. A negative value then means motion or acceleration in the opposite direction.
在一维问题中，选定正方向并始终一致使用。负值则表示运动或加速度沿相反方向。
2. The First Equation: v = u + at | 第一方程：v = u + at
The definition of acceleration gives a = (v − u) / t for uniform acceleration. Rearranging this definition gives the first equation of motion.
加速度的定义为匀加速时 a = (v − u) / t。整理该定义即可得到第一个运动学方程。
v = u + at
This equation is useful when time is given or required, and acceleration is constant.
当已知或要求时间且加速度恒定时，这个方程很有用。
3. Deriving Displacement from Average Velocity | 由平均速度推导位移
For uniform acceleration, the velocity–time graph is a straight line, so the average velocity is exactly the mean of the initial and final velocities.
对于匀加速运动，速度-时间图是一条直线，因此平均速度恰好等于初速度与末速度的平均值。
Displacement is average velocity multiplied by time, giving s = ½(u + v)t.
位移等于平均速度乘以时间，因此得到 s = ½(u + v)t。
s = ½(u + v)t
4. The Second Equation: s = ut + ½at² | 第二方程：s = ut + ½at²
Substitute v = u + at into s = ½(u + v)t:
将 v = u + at 代入 s = ½(u + v)t：
s = ½(u + u + at)t = ½(2u + at)t = ut + ½at²
This form is useful when initial velocity, acceleration and time are known but final velocity is not needed. The term ½at² represents the extra displacement caused by acceleration.
当已知初速度、加速度和时间而不需要末速度时，这个形式很有用。项 ½at² 表示由加速度引起的额外位移。
5. The Third Equation: v² = u² + 2as | 第三方程：v² = u² + 2as
Eliminate time t from the first two equations. From v = u + at, t = (v − u) / a. Substitute into s = ½(u + v)t:
从前两个方程中消去时间 t。由 v = u + at 可得 t = (v − u) / a。将其代入 s = ½(u + v)t：
s = ½(u + v) × (v − u) / a = (v² − u²) / 2a ⇒ v² = u² + 2as
This equation is especially useful when time is not mentioned or required.
当题目没有给出或不需要时间时，这个方程尤其有用。
6. The Fourth Equation: s = ½(u + v)t | 第四方程：s = ½(u + v)t
The fourth equation has already been obtained from the average-velocity argument. It is often listed as s = ½(u + v)t.
第四个方程已经通过平均速度的推导得到。它通常写作 s = ½(u + v)t。
s = ½(u + v)t
This equation does not contain acceleration, so use it when acceleration is not given or not required.
该方程不含加速度，因此当未给出或不需要加速度时可使用它。
7. The Fifth Equation: s = vt − ½at² | 第五方程：s = vt − ½at²
Rearrange v = u + at to express u = v − at, then substitute into s = ½(u + v)t:
整理 v = u + at 得到 u = v − at，然后代入 s = ½(u + v)t：
s = ½((v − at) + v)t = ½(2v − at)t = vt − ½at²
This form is less commonly used but can help when final velocity, acceleration and time are known.
这种形式较少使用，但当已知末速度、加速度和时间时会很方便。
8. Graphical Derivation Using Velocity–Time Graphs | 利用速度-时间图推导
The equations can also be derived from a velocity–time graph. Acceleration is the gradient of the line, so a = (v − u) / t. Displacement is the area under the line.
方程也可以从速度-时间图推导。加速度是直线的斜率，所以 a = (v − u) / t。位移是直线下方的面积。
Split the area into a rectangle and a triangle. The rectangle has area ut, and the triangle has area ½(v − u)t. Since v − u = at, the total area is s = ut + ½at².
将面积分成一个矩形和一个三角形。矩形的面积为 ut，三角形的面积为 ½(v − u)t。由于 v − u = at，总面积为 s = ut + ½at²。
The same area can be written as the trapezium area s = ½(u + v)t, which gives the average-velocity equation.
同一面积也可写成梯形面积 s = ½(u + v)t，从而得到平均速度方程。
9. Signs, Vector Nature and Problem Tips | 符号、矢量性与解题建议
All five equations are vector equations. Choose a positive direction, for example upward or to the right. A negative value of u, v, a or s then means the opposite direction.
所有五个方程都是矢量方程。选定正方向，例如向上或向右。u、v、a 或 s 为负值表示沿相反方向。
Always check that acceleration is constant before using the equations. If acceleration changes, split the motion into stages where it is constant.
在使用这些方程前，始终检查加速度是否恒定。如果加速度改变，将运动分成加速度恒定的阶段。
Use SI units and keep signs consistent throughout the calculation.
使用国际单位制，并在整个计算过程中保持符号一致。
10. Worked Example | 例题解析
A car accelerates uniformly from rest at 2.0 m s⁻² for 5.0 s. Calculate its final velocity and displacement.
一辆汽车从静止开始以 2.0 m s⁻² 匀加速，持续 5.0 s。计算其末速度和位移。
Using v = u + at: u = 0, a = 2.0 m s⁻², t = 5.0 s, so v = 0 + 2.0 × 5.0 = 10 m s⁻¹.
使用 v = u + at：u = 0，a = 2.0 m s⁻²，t = 5.0 s，因此 v = 0 + 2.0 × 5.0 = 10 m s⁻¹。
Using s = ut + ½at²: s = 0 × 5.0 + 0.5 × 2.0 × (5.0)² = 25 m. Alternatively, s = ½(u + v)t = 0.5 × (0 + 10) × 5.0 = 25 m.
使用 s = ut + ½at²：s = 0 × 5.0 + 0.5 × 2.0 × (5.0)² = 25 m。或者用 s = ½(u + v)t = 0.5 × (0 + 10) × 5.0 = 25 m。
11. Common Misconceptions | 常见误区
Students often forget that s is displacement, not distance. If an object changes direction, distance and displacement can be different.
学生经常忘记 s 是位移而不是路程。如果物体改变方向，路程和位移可能不同。
The equation v = s / t gives average velocity in general, but only for uniform motion is it equal to the instantaneous velocity. Do not mix it with the SUVAT equations.
方程 v = s / t 一般给出平均速度，但只有在匀速运动中才等于瞬时速度。不要把它与 SUVAT 方程混淆。
The average velocity (u + v) / 2 is only valid when acceleration is constant. Do not use it for variable acceleration.
平均速度 (u + v) / 2 只在加速度恒定时成立。加速度变化时不要使用它。
Unit errors are common: remember that acceleration is in m s⁻², and time must be in seconds.
单位错误很常见：记住加速度的单位是 m s⁻²，时间必须用秒。
12. Summary and Exam Strategy | 总结与备考策略
List the five equations together: v = u + at, s = ½(u + v)t, s = ut + ½at², s = vt − ½at², v² = u² + 2as. Each involves four of the five quantities; choose the equation that includes the three knowns and the one unknown.
将这五个方程列在一起：v = u + at、s =
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply