How do you derive the equations of motion?

The equations of motion are derived using the principles of calculus and Newton's laws of motion.

In more detail, the equations of motion describe the relationship between displacement, velocity, acceleration, and time. They are fundamental to the study of physics and are derived from Newton's laws of motion using calculus.

The first equation of motion, v = u + at, is derived from the definition of acceleration. Acceleration is the rate of change of velocity with respect to time. If we let u be the initial velocity, v the final velocity, a the acceleration and t the time, then the change in velocity is v - u, and the change in time is t. Therefore, acceleration a = (v - u) / t, which rearranges to give the first equation of motion.

The second equation of motion, s = ut + 0.5at^2, is derived from the definition of velocity, which is the rate of change of displacement with respect to time. If we let s be the displacement, then the change in displacement is s, and the change in time is t. Therefore, velocity v = s / t. Substituting the first equation of motion into this gives s = ut + 0.5at^2.

The third equation of motion, v^2 = u^2 + 2as, is derived by eliminating t between the first two equations. This gives v^2 = u^2 + 2as.

These equations are very useful in physics because they allow us to predict the future state of a moving object if we know its current state and the forces acting on it. They are also the starting point for more advanced topics in physics, such as the study of energy and momentum.