Kinematics

Kinematics is about the calculations of motion. The topic is introduced in High School Physics and typically three equations are shown.

v = v_0 + at

x = x_0 + v_0 t + \dfrac {1}{2} a t^2

We can get to these equations using calculus. Arguably, the desire to understand these equations is what led to the development of calculus.

The third equation, shown below, does not have time in it. We can (and should) show you how we get to it.

v_f^2 = v_i^2 + 2a(x_f - x_i)

Appendix A

v = \int_{t_i}^{t_f}a dt

\int_{t_i}^{t_f} \: a dt = a \int_{t_i}^{t_f} \: dt = a |_{t_i}^{t_f} \: t = a(t_f - t_i)

v= a(t_f - t_i)

It is standad practice to define t by t=t_f - t_i and if we do this then v = at.

Appendix B

x_f = x_i + v_i t + \dfrac {1}{2} a t^2

x_f - x_i = v_i t + \dfrac {1}{2} a t^2

Our goal us to remove ‘t’ from the above equation. We can do that using

v_f - v_i = at

t = \dfrac {v_f - v_i} {a}

t^2 = \dfrac {v_f^2 - 2v_fv_i + v_i^2}{a^2}

x_f - x_i = v_i t + \dfrac {1}{2} a t^2

x_f - x_i = v_i \dfrac {v_f - v_i} {a} + \dfrac {1}{2} a \dfrac {v_f^2 - 2v_fv_i + v_i^2}{a^2}

x_f - x_i = \dfrac {v_fv_i - v_i^2} {a} + \dfrac {v_f^2 - 2v_fv_i + v_i^2}{2 a}

x_f - x_i = \dfrac {2v_fv_i - 2v_i^2} {2a} + \dfrac {v_f^2 - 2v_fv_i + v_i^2} {a}

x_f - x_i = \dfrac {2v_fv_i - 2v_i^2 +v_f^2 - 2v_fv_i + v_i^2} {2a}

x_f - x_i = \dfrac { - 2v_i^2 +v_f^2 + v_i^2} {2a}

x_f - x_i = \dfrac { v_f^2 - v_i^2} {2a}

2a(x_f - x_i) = v_f^2 - v_i^2

Appendix D

You might find the following fourth equation listed:

x_f - x_i = \dfrac {v_i + v_f}{2} t

The above equation gets change of distance by multiplying average velocity against Time.

Appendix E

If you read mukriple authors, you will notice that different choices are made for notation.

d = x_f - x_i

Someone else uses the following:

d = x - x_0

We would like for you to believe that our choice and I and F for initial and Final is more sensible. We would be pleased if you told us that you kind of thought someone else’s choice to have a subscript on one of the two but not the other was an inconsistency…

However, we could argue that there are merits to working with \Delta t = t - t_0. This choice emphasizes t as being a variable and t_0 as being an initial condition.

We would use ‘f’ and ‘i’ to replace ‘a’ and ‘b’ in several definition