1
CHAPTER 7
PROJECTILES
7.1
No Air Resistance
We suppose that a particle is projected from a point O at the origin of a coordinate system,
the
y
axis being vertical and the
x
axis directed along the ground.
The particle is projected in
the
xy
plane, with initial speed
V
0
at an angle
α
to the horizon.
At any subsequent time in its
motion its speed is
V
and the angle that its motion makes with the horizontal is
ψ
.
The initial horizontal component if the velocity is
V
0
cos
α
, and, in the absence of air
resistance, this horizontal component remains constant throughout the motion.
I shall also
refer to this constant horizontal component of the velocity as
u
.
I.e.
u
=
V
0
cos
α
=
constant
throughout the motion.
The initial vertical component of the velocity is
V
0
sin
α
, but the vertical component of the
motion is decelerated at a constant rate
g
.
At a later time during the motion, the vertical
component of the velocity is
V
sin
ψ
, which I shall also refer to as
v
.
In the following, I write in the left hand column the horizontal component of the equation of
motion and the first and second time integrals; in the right hand column I do the same for the
vertical component.
Horizontal.
Vertical
0
=
x
&
&
g
y

=
&
&
7.1.1
a
,
b
α
=
=
cos
0
V
u
x
&
gt
V
y

α
=
=
sin
0
v
&
7.1.2
a
,
b
x
V t
=
0
cos
α
y
V t
gt
=

0
1
2
2
sin
α
7.1.3
a
,
b
The two equations 7.1.3
a
,
b
are the parametric equations to the trajectory. In vector form,
these two equations could be written as a single vector equation:
r
V
g
0
=
+
t
t
1
2
2
.
7.1.4
Note the + sign on the right hand side of equation 7.1.4.
The vector
g
is directed downwards.
The
xy
equation to the trajectory is found by eliminating
t
between equations 7.1.3
a
and
7.1.3
b
to yield:
y
x
gx
V
=

tan
cos
.
α
α
2
0
2
2
2
7.1.5
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document