Question

A particle moves along the x-axis so that at any time t, measured in seconds, its position is given by s(t) = 5cos(t) − sin(3t), measured in feet. What is the acceleration of the particle at time t = π seconds?

Answer 1

Let's find the derivative of this equation:
s(t) = 5cos(t) - sin(3t)
s'(t) = 5*(-sin(t)) - 3*cos(3t)
s'(t) = -5sin(t) - 3*cos(3t)
This tells us the velocity of the partical at any time t. To find the function for acceleration, we derive one more time:
s'(t) = -5sin(t) - 3*cos(3t)
s''(t) = -5cos(t) - (-9*sin(3t))
s''(t) = -5cos(t) + 9sin(3t)
Now we just plug in π for s''(t) to find our acceleration at time t
s''(π) = -5cos(π) + 9sin(3π)
s''(π ) = -5*-1 + 9 *(0)
s''(π ) = 5
The acceleration is 5 at time t=π