Preparation for Physics
V-65
1-D Equations of Motion for Constant Acceleration
The definitions of average velocity and average acceleration may be used, along with special
conditions imposed when motion occurs with constant acceleration, to derive equations that
describe the motion of an object at different times.
The definitions of average velocity and average acceleration are
vav ≡
Δx x f − xi
=
Δt
t f − ti
and
aav ≡
Δv v f − vi
=
Δt t f − ti
Two special conditions that exist when motion occurs with constant acceleration are
(1) v =
av
1
2
(v + v )
i
f
and
(2) aav = a
Exercise: Use the above four relationships to derive the equations of motion for constant
acceleration:
(0) Δx =
1
2
(v + v )Δt
i
(2) v = v + a Δt
f
i
f
(3) v 2 = v 2 + 2aΔx
(1) Δx = v Δt + 1 a (Δt )2
f
i
i
2
Frequently these will be written assuming ti = 0 and t f = t so that Δt = t f − t i = t f = t. .
Then these equations may be written:
(v + v ) t
(0)
x f = xi +
(1)
x f = xi + vi t + 12 at 2
1
2
i
f
(2) v = v + a t
f
i
(3) v 2 = v 2 + 2a (x − x )
f
i
f
i
Other notation frequently used in physics texts sets xf = x, xi = x0, vf = v and vi = v0. Using this
notation, x0 and v0 are the position and velocity at time t0 = 0 and x and v are the position and
velocity at some later time t.
Then the 1-D equations of motion for constant acceleration are written:
(v0 + v ) t
(0)
x = x0 +
(1)
x = x0 + v0 t + 12 at 2
1
2
(2) v = v + a t
0
(3) v 2 = v 2 + 2a (x − x ) .
0
0
We can use these equations to solve kinematics problems algebraically.
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