Deriving All Key Kinematic Equations [closed]

Question

I would like to know how to derive all the key kinematic equations, I currently know how to derive about 3 and would like help with the other two.

$v_f = v_i+at$

This seems pretty simple as $at = ∆x$ and $v_f = v_i+∆x$

$∆x = \frac{v_i+v_f}{2}t$

I know this is derived from $∆x = Vavg(t)$ (and Vavg = $\frac{v_i+v_f}{2}$)

$∆x = v_it+\frac{1}{2}at^2$

This would be a further derivation of $∆x = \frac{v_i+v_f}{2}t$ as $v_f = v_i+at$ so $∆x = \frac{v_i+v_f}{2}t$ would be $∆x = \frac{v_i+v_i+at}{2}t$ which can be simplified to $∆x = \frac{1}{2}at^2$

$∆x = v_ft-\frac{1}{2}at^2$ (this is unknown to me)

Would this also be connected to $∆x = \frac{v_i+v_f}{2}t$?

$v_f^2=v_i^2+2a∆x$ (this is unkown to me)

I have no idea

P.S If possible, could you please try and keep the derivations calculus free.

Answer 1

Well done on understanding $3$ out of $5$. Firstly, your third equation implies your fourth because $$\Delta x = (v_f - at)t+\frac{1}{2}at^2=v_f-\frac{1}{2}at^2.$$ Secondly, $$v_f^2-v_i^2=(v_f-v_i)(v_f+v_i)=at\frac{2\Delta x}{t}=2a\Delta x,$$so $v_f^2=v_i^2+2a\Delta x$.