Expert: Ahmed Salami - 4/1/2009

Question
A particle moves on the x-axis so that at any time t its velocity v(t) = sin 2t subject to the condition x(0) = 0 where x(t) is the position function.  Which of the following is an expression for x(t)?

a. cost 2t + 1/2
b. -1/2sint 2t + 1/2
c. -1/2cost 2t
d. -1/2cos 2t + 1/2
e. -1/2cost2t - 1/2

How would i go about solving this problem?

Answer
Hi Davis,
v(t) = sin 2t
To get x(t) from this, we integrate. We do this because v = dx/dt
dx = vdt
x = {vdt  [{ represents the integral sign]
 = {sin 2t dt
 = (-1/2)cos 2t + C
but,
x(0) = 0
i.e at t = 0, x = 0
From x = (-1/2)cos 2t + C
0 = (-1/2)cos 2(0) + C
0 = (-1/2).1 + C
0 = -1/2 + C
C = 1/2
Therefore,
x = (-1/2)cos 2t + 1/2    (option d)

Regards