Kinematics describes a particle moving in a straight line using functions of time \(t\). You already differentiate and integrate; here both are applied to motion: differentiating moves from displacement towards acceleration, and integrating moves back. Direction along the line is encoded by sign.
Definitions
| Quantity | Meaning |
|---|---|
| Displacement \(s\) | Change in position from a fixed point \(O\). Its magnitude is the distance from \(O\). |
| Velocity \(v\) | Rate of change of displacement: \(v=\dfrac{\mathrm{d}s}{\mathrm{d}t}\). Its magnitude is the speed. |
| Acceleration \(a\) | Rate of change of velocity: \(a=\dfrac{\mathrm{d}v}{\mathrm{d}t}=\dfrac{\mathrm{d}^2s}{\mathrm{d}t^2}\). |
The differentiate-integrate chain
\(s\ \xrightarrow{\ \text{differentiate}\ }\ v\ \xrightarrow{\ \text{differentiate}\ }\ a\)
\(s\ \xleftarrow{\ \text{integrate}\ }\ v\ \xleftarrow{\ \text{integrate}\ }\ a\)
| Direction | Relations |
|---|---|
| Differentiate | \(v=\dfrac{\mathrm{d}s}{\mathrm{d}t}\) \(a=\dfrac{\mathrm{d}v}{\mathrm{d}t}=\dfrac{\mathrm{d}^2s}{\mathrm{d}t^2}\) |
| Integrate | \(v=\displaystyle\int a\,\mathrm{d}t\) \(s=\displaystyle\int v\,\mathrm{d}t\) |
Each integration produces an arbitrary constant. Substitute the given information — such as the initial velocity or the starting position — to find it.
Assumptions at \(t=0\)
| Assumption | When it is allowed |
|---|---|
| \(s=0\) when \(t=0\) | Only when the origin is chosen at the starting point, or the question states that the particle starts at the fixed origin. |
| \(v=0\) when \(t=0\) | Only when the question states the particle starts from rest. |
| \(a=0\) when \(t=0\) | Only when the question gives it. |
Never assume \(v=0\) or \(a=0\) at \(t=0\). Every constant of integration must come from information given in the question.
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