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

Step 1 ：We are given the second order differential equation \(\frac{d^{2} s}{d t^{2}}=-4 \sin \left(2 t-\frac{\pi}{2}\right)\) 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: \(s(t) = \boxed{300t - \sin(2t)}\)