A mass attached to an extended spring is held at an initial displacement of x = 0.5 metres from a fixed point on a level surface

Question

The mass is then released and the spring allowed to contract freely with an oscillating motion along a straight line about the fixed point (x = 0). If the velocity of the motion is described by the formula v(t) = − π/ 4 sin( πt /2 ) metres per second , what is the formula for the displacement x(t)? In particular, what is the displacement when t = 2 seconds?

Answer 1

At t=0, x=1/2   and   v=dx/dt=-(π/4)sin(πt/2)

Integrate dx/dt=(-1/2)(π/2)sin(πt/2)    WRT t -->

x=(1/2)cos(πt/2) + c   but  c=0  since  x(0)=1/2 

so    x=(1/2)cos(πt/2)    and    x(2)=(1/2)cos(π)=-1/2