Mathematical Models of
Motion
Mathematical Models of Motion
 Position vs. Time Graphs (When and
Where)
 Using equation to find out When and
Where
V = Δd / Δt = df – di / tf – ti
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)
Velocity vs Time
 V = Δd / Δt , So
Δd = V Δt
 Notice that the area
under the curve is v x t
Velocity (m/s)
2.5
2
1.5
1
0.5
0
0
1
2
tim e (s)
3
Mathematical Models of Motion
Velocity vs Time
5
Velocity (m/s)
 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
4
3
2
1
0
0
1
2
tim e (s)
3
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 at2
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
vf2 = vi2 +2aΔd
Eqn 7