Displacement with distime: displacement s, duration (time interval) t, velocities v1 and v, acceleration a:
To prove: v = v1 + at
a = (v – v1) / t by definition
⇒ at = (v – v1) 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, lenticities u1 and u, relent b:
To prove: u = u1 + bs
b = (u – u1) / s by definition
⇒ bs = (u – u1) 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