Solve problems involving variable acceleration

Understanding Variable Acceleration

Variable acceleration refers to situations in which the acceleration of an object is not constant but changes with respect to time.
It’s important to understand the relationship between displacement, velocity and acceleration. Acceleration is the change in velocity over time, while velocity is the change in displacement over time.
Modelling variable acceleration often involves the use of calculus, as the acceleration function (usually given) is typically a derivative or an integral of the velocity or displacement functions respectively.

Use the kinematic equations when applicable. However, these equations assume constant acceleration, and should not be used when acceleration is variable.
Identify situations where acceleration is a function of time. In these cases, you may need to apply differential or integral calculus to find velocity and displacement.
When given an acceleration function of time, the velocity function can be obtained by integrating the acceleration function. Similarly, the displacement function can be obtained by integrating the velocity function.
Remember that each integration step will introduce an arbitrary constant. Utilise any initial conditions provided (such as initial velocity or initial position) to solve for these constants.

To solve problems, start by identifying the givens and what you need to find. If you are given an acceleration function and need to find a displacement function, you’ll need to integrate twice.
Carefully follow the units given in the problem and convert if necessary. Hence, the acceleration, velocity and displacement should be in appropriate corresponding units.
Initial conditions, if given, often represent values at time zero (such as an initial velocity or initial position). Use these to solve for the arbitrary constants that arise during integration.
Graphical understanding can also be beneficial. Knowing that integration of an acceleration-time graph gives a velocity-time graph and so on might be useful.
Take advantage of symmetries or zero points. For example, at the peak of a projectile’s motion, its vertical velocity component is zero. This can simplify calculations at times.
After obtaining expressions for acceleration, velocity or displacement as functions of time, properly evaluate them by substitifying relevant values of time to compute the required answers.
Always remember to check your answer by substituting it back into the original problem. If all specified conditions are satisfied, your answer is likely correct.