3 MOTION IN 3 DIMENSIONS 3.10 Motion with constant acceleration
0·Figure 14: Motion with constant velocityHence, the object’s velocity is given by
v(t) =drdt= v 0 + a t. (3.34)Note that dv/dt = a, as expected. In the above, the constant vectors r 0 and v 0
are the object’s displacement and velocity at time t = 0 , respectively.
As is easily demonstrated, the vector equivalents of Eqs. (2.11)–(2.13) are:s = v 0 t^ +^1
a t^2 , (3.35)
2
v = v 0 + a t, (3.36)
v^2 = v 2 + 2 a·s. (3.37)These equation fully characterize 3-dimensional motion with constant accelera-
tion. Here, s = r − r 0 is the net displacement of the object between times t = 0
and t.
The quantity as, appearing in Eq. (3.37), is termed the scalar product of vectorsa and s, and is defined
a·s = ax sx + ay sy + az sz. (3.38)The above formula has a simple geometric interpretation, which is illustrated in
Fig. 15. If |a| is the magnitude (or length) of vector a, |s| is the magnitude of
v
t = t
trajectoryt = 0
rr 0