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Non-uniform acceleration
Intermediate
When a varies, integrate calculus-style to get v and x.
v = ∫a dtWhen a changes with time, the constant-acceleration formulas fail. Instead integrate: v = v₀ + ∫a dt, x = x₀ + ∫v dt, or read areas on graphs.
If a = 6t, then v = 3t² and x = t³ (from rest at origin).
Never apply s = ut + ½at² to varying acceleration.
In more detail
When acceleration is not constant, the standard equations do not apply directly. Calculus, using derivatives and integrals, gives velocity and position from a changing acceleration.
Real-world uses
- Rocket engines and jet aircraft have acceleration that changes as fuel burns.
- Engineers model lift and drag forces that vary with speed.
- A rocket's acceleration changes as fuel burns, so engineers integrate the acceleration over time to find its speed.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
Can I still use kinematic equations?
Only over intervals where acceleration is approximately constant.
How do I get velocity from acceleration?
Integrate acceleration with respect to time, adding the initial velocity as a constant.
When can I not use the constant-acceleration equations?
When the acceleration changes during the motion. Then integrate the acceleration to get velocity, and integrate velocity to get position.
Related topics in Kinematics
Relative velocity in 1D · Stopping distance · Reaction time and braking · Motion graphs from calculus · Terminal velocity · Two-dimensional motion