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Lecture Example 1: Curvilinear motion
• Consider the function
• What is the acceleration along the path of travel at time, t=1sec?
jia
jiv
jir
)4( )12()(
)4( )14()(
)2( )()(
2
3
24
tt
ttt
tttt
Derivation of the normal and tangential components of acceleration.
RuleProduct dt
dv
dt
dv
vdt
d
dt
d
v
tt
t
t
uua
uv
a
uv
Lecture Example 2: Find the equation of the path, y=f(x). Find the normal and tangential component of the acceleration at t=0.25s.
Lecture Example 3: At time, t = π seconds determine the velocity of the particle, and determine the tangential and normal components of the acceleration. Draw and label the acceleration vectors on the graph below. The position of the particle is expressed in meters. Trigonometric functions should be evaluated in radians.
-4
-3
-2
-1
0
1
2
3
4
5
6
-6 -5 -4 -3 -2 -1 0 1 2 3 4
tdt
dtt
dt
dttt
dt
dsin sinsin
r(t) = (t sin t) i + (t cos t ) j 0 < t < 2 π sec
Spiral of Archimedes
Lecture Example 4: The jet plane travels along the vertical parabolic path. When at Point A it has reached a speed of 200 m/s which is increasing at a rate of 0.8 m/s2. Determine the magnitude of the acceleration of the plane when it is at Point A.
Note: The positions x and y are given in kilometers.