Mathematical Models of Motion

 Position vs. Time Graphs (When and Where)  Using equation to find out When and Where  V = Δd / Δt = d f – d i / t f – t iEqn 1  If we solve for “d f ” we get  d f = d i + vt Eqn 2

Mathematical Models of Motion  Velocity vs. Time Graphs  a = Δv / Δt = v f – v i / t f – t iEqn 3  If we solve for “v f ” we get  v f = v i + 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 = v i t + ½ (v f - v i )t  When the terms are combined (factored) you get…  Δd = ½ (v f + v i )t Eqn 5 OR d f = d i + ½ (v f + v i )t

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

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