\(\text{Gradient of}\hspace{0.5em}s\text{–}t=v\qquad\text{Gradient of}\hspace{0.5em}v\text{–}t=a\)
\(s(t_2)-s(t_1)=\int_{t_1}^{t_2}v\,\mathrm dt\)
Displacement-time graph: gradient \(=\) velocity.
Velocity-time graph: gradient \(=\) acceleration; signed area under graph \(=\) change in displacement.
Displacement–time graph
| Interval | Velocity from the gradient |
|---|---|
| \(0<t<2\) | |
| \(2<t<5\) | |
| \(5<t<8\) |
Velocity–time graph
| Quantity | Working |
|---|---|
| Acceleration, \(0<t<2\) | |
| Acceleration, \(2<t<5\) | |
| Acceleration, \(5<t<8\) | |
| Change in displacement, \(0\le t\le 8\) | |
| Total distance, \(0\le t\le 8\) |
Below the time axis, area contributes negative displacement but positive distance. Position relative to the origin also requires the starting position.
A two-sided derivative does not exist at a corner; an endpoint may have a one-sided rate.
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