Mathematical Models of Motion Mathematical Models of Motion

Mathematical Models of Motion

Mathematical Models of Motion § Position vs. Time Graphs (When and Where) § Using equation to find out When and Where § V = Δ d / Δ t = df – di / t f – t i Eqn 1 § If we solve for “df” we get § df = di + vt Eqn 2

Mathematical Models of Motion § Velocity vs. Time Graphs § a = Δv / Δt = vf – vi / tf – ti Eqn 3 § If we solve for “vf” we get § vf = vi + at Eqn 4

Mathematical Models of Motion § Area under the curve of a V vs. T graph § (Length x width) or (velocity x time) § V = Δd / Δt , So Δd = V Δt § Notice that the area under the curve is v x t

Mathematical Models of Motion § Area under the curve for constant acceleration § Δd = vit + ½ (vf - vi)t § When the terms are combined (factored) you get… § Δd = ½ (vf + vi)t Eqn 5 OR df = di + ½ (vf + vi)t

Mathematical Models of Motion § Frequently, the final velocity at time “t” is not known § b/c vf = vi + at (eqn 4), and Δd = ½ (vf + vi)t (eqn 5) § We can substitute vf from the first equation (vf = vi + at) into the second equation (Δd = ½ (vf + vi)t ) § When we do, we get Δd = ½ ( vi + at + vi)t § OR Δd = vit + ½ at 2 Eqn 6

Mathematical Models of Motion § Sometimes “t” is not known, if we combine Δd = ½ (vf + vi)t (eqn 5) and vf = vi + at (eqn 4), we can eliminate the variable “t” § solving (vf = vi + at) for “t” t = (vf – vi) / a § Substitute (vf – vi) / a in for “t” in equation 4 and you get Δd = ½ (vf + vi) (vf – vi) / a § Foil and solve for “vf” and you get vf 2 = vi 2 +2 aΔd Eqn 7