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If acceleration is not constant, we cannot use a single equation to describe the motion of an object. Instead, we need to use calculus and consider the varying acceleration as a function of time.

In such cases, we can use calculus-based equations of motion, such as those derived from Newton's laws of motion. One common equation is:

v = u + ∫(a(t) dt)

where:

  • v is the final velocity of the object,
  • u is the initial velocity of the object,
  • a(t) is the acceleration as a function of time, and
  • ∫(a(t) dt) represents the integral of the acceleration over time, which gives the change in velocity.

This equation accounts for the fact that acceleration can vary with time by integrating the acceleration function over the relevant time interval. The integral of the acceleration function gives the net change in velocity during that time interval.

It's important to note that for complex or non-uniform acceleration functions, solving these equations may require advanced mathematical techniques and numerical methods.

Additionally, it's worth mentioning that there are specific equations of motion for certain types of acceleration functions, such as uniformly accelerated motion (constant acceleration) or motion under simple harmonic motion, which have simplified equations derived for those specific cases.

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