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14 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Equations for Ch.3 Kinematics in 2
-
D


For X components

a
x

= 0

For Y components

a
y

= g =
-
9.8 m/s
2


v
ix

= v
fx

= v
x




v
fy
= v
iy

+
g
.t

d
x

= v
x

. t


d
y

= (v
iy
+ v
fy
).t / 2

---------------------




d
y

= v
iy
.t +
½
g
.t
2


v
x

= V cos






v
fy
2

= v
iy
2

+ 2.
g
.d
y

v
Y

= V sin




=

tan
-
1

(v
y

/ v
x
)


v =
2 2
x y
v v


Use the absolute value of g as 9.8 m/s
2
, in 4 equations

below
; That is, time
to apex, time of flight, maximum height and range of a projectile.


Time to the apex





sin
i
a
v
t
g




Time of flight (hang time)




2 sin
i
f
v
t
g




Maximum height of a projectile





2
2
2
sin
sin
2 2
i
i
v
v
h
g g


 


Range of a projectile





2
sin 2
i
v
R
g




For the two equations below, use g

=
-
9.8 m/s
2
.
The below two equations
are only for vertical components.
d
y

and velocity =
v
fy
,
and are valid only
if initial vertical velocity is zero,
v
iy

= 0

Time for a particle to fall height
d
y



2
Y
d
t
g


Velocity of a particle after falling height
d
y


v
fy


2
Y
gd


=
gt



Ch 3 Handouts

Kinetmatics in 2
-
D


Projectile motion is kinematics in 2
-
D, when the object only experiences acceleration (
a
Y

=
-
9.8 m/sec
2
) due to gravity. Neglecting air resistance,
a
X

= 0
, the horizontal component
of the velocity
v
X

would never change. Only
v
Y

is affected by gravity and continuously
decreases
on the way up reaching zero
Y
-
velocity at the top, and then increasing in the
downward direction.
However remember that X
-
velocity is still the same, so the resultant
velocity is the Pythagorean sum of the X & Y velocities.
The magnitudes of the Y
-
velocities are the same at equal hei
ghts and opposite in directions. Time only depends
on the Y
-
components of motion. The motion of a projectile can be analyzed by treating
the X and Y components of the vector separately and realizing that time ‘t’ is the same
for both
.

The equations of kine
matics are written only for the Y
-
components. There is
only one equation for X
-
components as there is no acceleration for X
-
direction.


If a projectile is thrown with initial velocity
v
i
and angle

t桥n

v
X

= v
i

. cos


v
iY

= v
i

. sin



At the top
v
FY

= 0, v
X

does not change during the entire flight.

Remember that X & Y components of motion are completely independent of each other,
and time is the only variable that can be shared between the two.


The time to apex, time of flight, Range and maximum he
ight equations are only valid for
a full range
.
Below are examples where those equations
cannot
be used.