Linear Motion with Variable Acceleration

Basic Concepts

Linear motion refers to the motion of an object along a straight line.
Variable acceleration signifies that the rate of change of velocity is not constant.
This concept ties into the fundaments of calculus and requires the use of integration and differentiation.

The change in velocity, also known as acceleration, is the derivative of velocity with respect to time (a = dv/dt).
Variable acceleration means this rate of change is not constant, therefore the acceleration function will be a function of time, a = a(t).
The change in position, or displacement, is the integral of velocity over time.
Velocity can be expressed as the integral of acceleration over time plus an initial velocity (v = ∫a dt + u), where ‘u’ is the initial velocity (at t = 0).

By integrating the acceleration function, you can obtain the velocity-time equation.
Similarly, by integrating the velocity function, you can obtain the displacement-time equation.
These equations can then be used to find out the final position, velocity or acceleration at any given time.

Write down the given acceleration function, initial velocity and initial position.
Integrate the acceleration function to find the velocity function. Remember to include the ‘plus C’ constant of integration, which can be determined using initial conditions.
Integrate the velocity function to find the displacement function.
Use these equations to find the values of your position, velocity, and acceleration at any desired time.

Understanding linear motion with variable acceleration is useful in physics, engineering and any field where modelling the movement of objects is important.