2.4. CURVILINEAR MOTION OF PARTICLES 19
is gives t D 14:4925 s or t D 2:8107 s. us, t D 14:4925 s is chosen and substituting
this value into Equation (2.14d) to give the horizontal distance
x D 1187:1560 m:
(b) When the missile reaches its highest elevation the vertical velocity is zero. at is,
0 D 57:3577
2
19:62y ) y D
57:3577
2
19:62
m D 167:6812 m:
erefore, the highest elevation achieved above the ground is 367.6812 m.
2.4 CURVILINEAR MOTION OF PARTICLES
Aside from rectilinear motion, a particle can undergo curvilinear motion in the physical world.
e curvilinear motion of a particle is the motion of that particle along a curved path or trajec-
tory. With reference to Figure 2.8, the particle P has a position vector
*
r and the velocity
*
v of
the particle is defined as
*
v D
d
*
r
dt
: (2.15)
at is, the velocity is a vector tangent to the path of the particle and has a magnitude v (also called
the speed) which is the time derivative of the length s of the arc described by the particle
v D lim
t !0
s
t
D
ds
dt
: (2.16)
e acceleration
*
a
of a particle is thus defined as
*
a D
d
*
v
dt
: (2.17)
Path
y
z
x
r⃑
Δ r
⃑
Δs
r⃑'
P'
P
O
υ⃑
Figure 2.8: Position and velocity vectors of a particle in a curved path.
20 2. KINEMATICS OF PARTICLES
Path
Ho
dograph
y
z
x
r⃑
Δ r⃑
Δs
r⃑'
P'
P
O
a⃑
⃑'
⃑
Δ⃑
Figure 2.9: Illustration of acceleration of a particle.
In general, the acceleration of a particle in a curved path is not a vector tangent to the path of the
particle. is point is illustrated in Figure 2.9. Note that the curve traced by the tip of velocity
*
v is known as the hodograph of the motion.
For two particles A and B travelling in space as shown in Figure 2.10 the
relative position
vector of B w.r.t. A,
*
r
B=A
; may be written as
*
r
B
D
*
r
A
C
*
r
B=A
: (2.18)
Similarly, the relative velocity of B w.r.t. A,
*
v
B=A
; may be written as
*
v
B
D
*
v
A
C
*
v
B=A
(2.19)
and the relative acceleration of B w.r.t. A,
*
a
B=A
; may be written as
*
a
B
D
*
a
A
C
*
a
B=A
: (2.20)
Y
y
B
Z
z
X
x
r⃑
B
r⃑
B/A
r⃑
A
A
O
Figure 2.10: Relative motion of two particles A and B.
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