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Calculus in kinematics
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Displacement, velocity and acceleration are linked by differentiation and integration.
The chain of links
Differentiate to go down: s → v → a
v = dsdt, a = dvdt
Integrate to go up: a → v → s
v = dsdt, a = dvdt
Integrate to go up: a → v → s
At rest and maximum speed
A particle is at rest when v = 0.
Velocity is greatest or least when a = 0.
Velocity is greatest or least when a = 0.
Using conditions
When you integrate, add a constant and find it from a given value.
a = 4 − 2t, v = 5 when t = 0: v = 4t − t2 + c, and c = 5.
a = 4 − 2t, v = 5 when t = 0: v = 4t − t2 + c, and c = 5.
s = t3 − 6t2 + 9t. Find when the particle is at rest.
- v = 3t2 − 12t + 9
- 3(t2 − 4t + 3) = 0
- 3(t − 1)(t − 3) = 0
Answer: t = 1 and t = 3
v = dsdt, a = dvdt. Integrate to reverse, and use given values to find the constant. At rest: v = 0.
The interactive lesson includes the diagrams for this topic.
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