Variable Acceleration

This lesson covers motion with variable acceleration, where acceleration changes over time rather than remaining constant. In these situations, the SUVAT equations cannot be used, and instead we use calculus (differentiation and integration) to relate displacement, velocity, and acceleration.

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The Calculus Relationships

Relationship	Calculus operation
Velocity from displacement	$v = \frac{ds}{dt}$v=dtds​ (differentiate $s$s with respect to $t$t)
Acceleration from velocity	$a = \frac{dv}{dt}$a=dtdv​ (differentiate $v$v with respect to $t$t)
Acceleration from displacement	$a = \frac{d^2s}{dt^2}$a=dt2d2s​ (differentiate $s$s twice)
Displacement from velocity	$s = \int v \, dt$s=∫vdt (integrate $v$v with respect to $t$t)
Velocity from acceleration	$v = \int a \, dt$v=∫adt (integrate $a$a with respect to $t$t)

Exam Tip: When the question says acceleration is "variable" or gives displacement/velocity as a function of time (e.g., $s = 3t^2 - 2t + 1$s=3t2−2t+1), you must use calculus. SUVAT equations only work for constant acceleration.

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Differentiation: Finding Velocity and Acceleration

Example: A particle moves along a straight line with displacement $s = 2t^3 - 9t^2 + 12t$s=2t3−9t2+12t metres, where $t$t is in seconds.