Mathematical Models of Mathematical Models of MotionMotion

Mathematical Models of MotionMathematical Models of Motion

Position vs. Time Graphs (When and Position vs. Time Graphs (When and Where)Where)

Using equation to find out When and Using equation to find out When and WhereWhere

V = V = ΔΔd / d / ΔΔt = dt = dff – d – dii / t / tff – t – tii Eqn 1Eqn 1

If we solve for “If we solve for “ddff” we get” we get

ddff = d = dii + vt + vt Eqn 2Eqn 2

Mathematical Models of MotionMathematical Models of Motion

Velocity vs. Time Graphs Velocity vs. Time Graphs a = a = ΔΔv / v / ΔΔt = vt = vff – v – vii / t / tff – t – tii Eqn 3Eqn 3

If we solve for “If we solve for “vvff” we get” we get

vvff = v = vii + at + at Eqn 4Eqn 4

Mathematical Models of MotionMathematical Models of Motion

Area under the curve Area under the curve of a V vs.T graphof a V vs.T graph

(Length x width) or (Length x width) or (velocity x time)(velocity x time)

V = V = ΔΔd / d / ΔΔt , So t , So ΔΔd = V d = V ΔΔt t

Notice that the area Notice that the area under the curve is v x tunder the curve is v x t

Velocity vs Time

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Mathematical Models of MotionMathematical Models of Motion

Area under the curve Area under the curve for constant for constant accelerationacceleration

ΔΔd = vd = viit + ½ (vt + ½ (vf f - v- vii)t)t

When the terms are When the terms are combined (factored) combined (factored) you get…you get…

ΔΔd = ½ (vd = ½ (vf f + v+ vii)t )t Eqn 5Eqn 5

OR dOR dff = d = dii + ½ (v + ½ (vf f + v+ vii)t)t

Velocity vs Time

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Mathematical Models of MotionMathematical Models of Motion

Frequently, the final velocity at time “t” is not Frequently, the final velocity at time “t” is not knownknown

b/c b/c vvff = v = vii + at + at (eqn 4), and (eqn 4), and ΔΔd = ½ (vd = ½ (vf f + v+ vii)t)t (eqn 5) (eqn 5)

We can substitute We can substitute vvf f from the first equation from the first equation

(v(vff = v = vii + at) into the second equation + at) into the second equation

((ΔΔd = ½ (vd = ½ (vf f + v+ vii)t ))t )

When we do, we get When we do, we get ΔΔd = ½ (d = ½ ( v vii + at + at + v+ vii)t )t

OR OR ΔΔd = d = vviit + t + ½ ½ atat22Eqn 6Eqn 6

Mathematical Models of MotionMathematical Models of Motion

Sometimes “t” is not known, if we combine Sometimes “t” is not known, if we combine ΔΔd = ½ (vd = ½ (vf f + v+ vii)t)t (eqn 5) and (eqn 5) and vvff = v = vii + at + at (eqn 4), (eqn 4),

we can eliminate the variable “t”we can eliminate the variable “t” solving (vsolving (vff = v = vii + at) for “t” + at) for “t” t = t = (v(vff – v – vii) / a) / a

Substitute Substitute (v(vff – v – vii) / a) / a in for “t” in equation 4 and in for “t” in equation 4 and

you get you get ΔΔd = ½ (vd = ½ (vf f + v+ vii) ) (v(vff – v – vii) / a) / a

Foil and solve for “vFoil and solve for “vff” and you get ” and you get vvff22

= v= vii22 +2a +2aΔΔdd Eqn 7Eqn 7