fundamentals of engineering supplied-reference handbook - Ventech!

Normal and Tangential Components
y
Unit vectors et and en are, respectively, tangent and normal
to the path with en pointing to the center of curvature. Thus
v = vt () e
t
2
t t n
a = at () e + ( v /ρ) e , where
ρ = instantaneous radius of curvature
Constant Acceleration
The equations for the velocity and displacement when
acceleration is a constant are given as
a(
t)
= a
0
v(
t)
= a ( t − t ) + v
0
s(
t)
= a ( t − t )
s
v
a
0
0
0
2
0
/ 2
+ v ( t − t ) + s ,
s = distance along the line of travel
0 0
0 0
0
0
= displacement at time t
0
0
0
where
v = velocity along the direction of travel
t
=
velocity at time t
= constant acceleration
t = time, and
= some initial time
For a free-falling body, a = g (downward) .
0
An additional equation for velocity as a function of position
may be written as
2
en
r
2
PATH
et
v = v0
+ 2a0 ( s − s0
)
x
30
y
θ
v0
x
DYNAMICS (continued)
For constant angular acceleration, the equations for angular
velocity and displacement are
α(
t)
= α0
ω(
t)
= α ( t − t ) + ω
0
θ(
t)
= α ( t − t )
θ= angular displacement
0
0
0
2
/ 2
θ = angular displacement at time t
0 0
ω= angular velocity
ω = angular velocity at time t
0 0
α = constant angular acceleration
t
0
t=
time, and
0
= some initial time
0
+ ω ( t − t ) + θ ,
0
0
0
where
An additional equation for angular velocity as a function of
angular position may be written as
ω
2
2
0
= ω
+ 2α0 ( θ − θ0
)
Non-constant Acceleration
When non-constant acceleration, a(t), is considered, the
equations for the velocity and displacement may be obtained
from
v(
t)
= ∫ a(
τ)
dτ
+ v
t
t
s(
t)
= ∫ v(
τ)
dτ
+ s
t
t
o
o
For variable angular acceleration
ω(
t)
= ∫ α(
τ)
dτ
+ ωt
t
t
θ(
t)
= ∫ ω(
τ)
dτ
+ θ
Projectile Motion
t
t
0
0
t
t
0
0
t
0
0