Kinematics, Non-Uniform Motion, Non-Constant Acceleration, 1D Motion - Physics (Mechanics)

Kinematics with non-constant acceleration requires the use of calculus, as standard algebraic kinematic equations only apply when acceleration is constant. By using integration and differentiation, you can relate position, velocity, and acceleration as functions of time, where velocity is the integral of acceleration, and acceleration is the derivative of velocity.

💡Key concepts
• Calculus is essential: Non-constant acceleration situations require calculus to solve, unlike constant acceleration problems that can be solved with algebra.
• Acceleration is a function: Acceleration (\(a(t)\)) is no longer a single value but a function of time, meaning it changes throughout the motion.
• Integration and differentiation:
⚬To find velocity (\(v(t)\)) from the acceleration function, you integrate \(a(t)\) with respect to time: \(v(t)=\int a(t)\,dt\).
⚬To find the position function (\(x(t)\)), you integrate the velocity function with respect to time: \(x(t)=\int v(t)\,dt\).
⚬Conversely, to check your work or find the acceleration function if you have position and velocity, you differentiate: \(a(t)=\frac{dv(t)}{dt}\) and \(v(t)=\frac{dx(t)}{dt}\).

💡Example
If an object's acceleration is given by the function \(a(t)=2t\,\text{m/s}^{2}\), you can find its velocity and position at any time \(t\):
• Find the velocity function:
\(v(t)=\int a(t)\,dt=\int 2t\,dt=t^{2}+C_{1}\)(where \(C_{1}\) is the constant of integration, determined by the initial velocity).
• Find the position function:
\(x(t)=\int v(t)\,dt=\int (t^{2}+C_{1})\,dt=\frac{t^{3}}{3}+C_{1}t+C_{2}\)(where \(C_{2}\) is the constant of integration, determined by the initial position).

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1FZHK2lvewIfauVNmVxQprxLvhBZfVv02/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1h8pFAHvhQo5qqSPsfYA9GETrdjMQg8z8/view?usp=drive_link

💡Chapters:
00:00 Non-uniform acceleration
01:30 Worked examples

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