Solve the following initial value problem.
$$\frac{d^{2}s}{d{t}^2}=-4\sin (2t-\frac{\pi }{2}),s^{\prime }(0)=300,s(0)=0$$

Step 1 ：We are given the second order differential equation $\frac{d^{2}s}{d{t}^2}=-4\sin (2t-\frac{\pi }{2})$ with initial conditions $s^{\prime }(0)=300$ and $s(0)=0$.

Step 2 ：The general approach to solve this kind of problem is to integrate the equation twice and then use the initial conditions to solve for the constants of integration.

Step 3 ：Integrating the equation once gives us $\frac{ds}{dt}=-4\cos (2t)+C1$, where $C1$ is the constant of integration.

Step 4 ：Using the initial condition $s^{\prime }(0)=300$, we can solve for $C1$ to get $C1=300$.

Step 5 ：Substituting $C1=300$ into the equation gives us $\frac{ds}{dt}=-4\cos (2t)+300$.

Step 6 ：Integrating the equation again gives us $s(t)=-\sin (2t)+300t+C2$, where $C2$ is another constant of integration.

Step 7 ：Using the initial condition $s(0)=0$, we can solve for $C2$ to get $C2=0$.

Step 8 ：Substituting $C2=0$ into the equation gives us the particular solution $s(t)=-\sin (2t)+300t$.

Step 9 ：Final Answer: $s(t)=\boxed{{\displaystyle 300t-\sin (2t)}}$