Kinematics: Variable Acceleration with Calculus

This lesson covers kinematics when acceleration is not constant, using differentiation and integration. This is a key part of the Edexcel 9MA0 specification that links pure mathematics with mechanics.

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When SUVAT Does Not Apply

The SUVAT equations require constant acceleration. When acceleration varies with time, we must use calculus.

The fundamental relationships are:

• Velocity is the derivative of displacement with respect to time: v = ds/dt
• Acceleration is the derivative of velocity with respect to time: a = dv/dt = d²s/dt²
• Displacement is the integral of velocity: s = ∫v dt
• Velocity is the integral of acceleration: v = ∫a dt

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Differentiation: From Displacement to Velocity to Acceleration

If the displacement is given as a function of time, s = f(t), then:

• v = ds/dt = f'(t)
• a = dv/dt = f''(t)

Example: A particle moves along a straight line with displacement s = 2t³ - 9t² + 12t, where s is in metres and t in seconds.

v = ds/dt = 6t² - 18t + 12 a = dv/dt = 12t - 18