iSoul In the beginning is reality.

# Kinematic derivations

Displacement with distime: displacement s, distime (time interval) t, velocities v1 and v, acceleration a:

To prove: v = v1 + at

a = (vv1) / t    by definition

at = (vv1)     multiply by t

v = v1 + at

To prove: s = v1t + ½ at2

vavg = s / t          by definition

vavg = (v1 + v) / 2        by definition

s / t = (v1 + v) / 2       combining these two

s = (v1 + v) t / 2         multiply by t

s = (v1 + (v1 + at)) t / 2       from above

s = (2v1 + at) t / 2

s = v1t + ½ at2

To prove: v2 = v12 + 2a·s

v = v1 + at         from above

v2 = (v1 + at)2         square

v2 = v12 + 2a·v1t + a2t2      expand

v2 = v12 + 2a·(v1t + ½ at2)       factor out 2a

v2 = v12 + 2a·s      result above

Dischronment with distance: distance s, dischronment t, legerities u1 and u, rapidation b:

To prove: u = u1 + bs

b = (uu1) / s   by definition

bs = (uu1)     multiply by s

u = u1 + bs

To prove: t = u1s + ½ bs2

uavg = t / s          by definition

uavg = (u1 + u) / 2       by definition

t / s = (u1 + u) / 2       combining these two

t = (u1 + u) s / 2         multiply by s

t = (u1 + (u1 + bs)) s / 2     result above

t = (2u1 + bs) s / 2

t = u1s + ½ bs2

To prove: u2 = u12 + 2b·t

u = u1 + bs        1st result above

u2 = (u1 + bs)2       square

u2 = u12 + 2b·u1t + b2s2     expand

u2 = u12 + 2b·(u1s + ½ bs2)     factor out 2b

u2 = u12 + 2b·t      2nd result above