Environmental Engineering Reference
In-Depth Information
Solution 1.3.3
The choice the co-ordinate system is up to us. Since we want to
separate the description of the motion into Cartesian components, we choose the
y-axis to be upwards and the x-axis to be horizontal and in the same plane as the
initial velocity. We then have
a
=−
g
j
,
and
u
=
u
x
i
+
u
y
j
=
u
cos
θ
i
+
u
sin
θ
j
.
We write the position of the projectile as
r
=
x
i
+
y
j
,
where x and y depend on time. For convenience we let
r
0
. Our choice
of co-ordinate system means that there is no acceleration in the x-direction. Thus
we have,
=
0
at t
=
x
=
u
x
t
=
ut
cos
θ.
In the y-direction we have
1
2
gt
2
.
These are parametric equations for x and y (with time as the parameter). To obtain
the path of the projectile we eliminate t to get y as a function of x:
1
2
gt
2
y
=
u
y
t
−
=
ut
sin
θ
−
u
y
u
x
g
2
u
x
g
2
u
2
cos
2
θ
x
2
.
x
2
y
=
x
−
=
x
tan
θ
−
This is the equation of a parabola (see Figure 1.8). To obtain the range of the
projectile we need to find values of x such that y
=
0
. These are x
=
0
, the launch
(u
2
sin 2
θ)/g, the horizontal range of the
projectile. Notice that the range is maximal for θ
(
2
u
2
sin
θ
cos
θ)/g
position, and x
=
=
45
◦
.
=
y (m)
140
120
100
80
60
40
20
x (m)
200
400
600
800
Figure 1.8 Parabolic path of a projectile fired at 30
◦
to the horizontal with an initial speed
of 105 ms
−
1
. Note that the distance scales are different on the horizontal and vertical axes.
