More generally, the equations of motion can be written as
x(t)=xi+vi\Delta t+12a(\Delta t)2
and
v(t)=vi+a\Delta t
\Delta t=t−ti
is the time that has elapsed since the beginning of the particle's motion. Here, t
is the current time and ti
is the time at which we start measuring the particle's motion. The terms xi
and vi
are, respectively, the position and velocity at t=ti
. As you can now see, the equations given at the beginning of this problem correspond to the case ti=0
, which is a convenient choice if there is only one particle of interest.
To illustrate the use of these more general equations, consider the motion of two particles, A and B. The position of particle A depends on time as xA(t)=xiA+viAt+(1/2)aAt2
. That is, particle A starts moving at time t=0
with initial velocity viA
from initial position xiA
. Suppose that at time t=t1
, particle B has twice the acceleration (aB=2aA)
, half the velocity that particle A had at time t=0
( viB=0.5viA
), and the same position that particle A had at time t=0
( xiB=xiA
).