A body is moving in simple harmonic motion with position function

Question

s(t) = sin t + cos t, with s in meters and t in seconds. Find the jerk at time t.

Answer 1

Jerk is another name for the 4th derivative, so we are looking for s''''(t) or s⁴(t)

Using the simple fact that the derivative of sin(t) is cos(t), and the derivative of cos(t) is -sin(t):

s(t) = sin(t) + cos(t)

s'(t) = cos(t) - sin(t)

s''(t) = -sin(t) - cos(t)

s'''(t) = -cos(t) + sin(t)

s''''(t) = sin(t) + cos(t)

So the jerk is sin(t) + cos(t)

Notice that our final answer is the same as the original question. The derivatives of sin and cos are cyclic, due to the function changing every time, and the sign swapping every other time. So every 4th derivative of either function will complete the cycle. So we would also expect the 8th, 12th, etc. derivatives to also be the same.