1. The problem statement, all variables and given/known data I'm given acceleration as a=-1.5*s where s is position, and I need to derive an expression for acceleration as a function of time. I am also given an initial velocity of 20 m/s and initial position of 0 m. 2. Relevant equations a=-1.5*s Characteristic Equation s(t)=C1*e-p*t*cos(sqrt(q)*t)+C2e-p*t*sin(sqrt(q)*t) 3. The attempt at a solution acceleration is the 2nd derivative of position, therefore a=-1.5*s is also equal to d2s/dt2=-1.5*s d2s/dt2+1.5*s=0 I took the laplace transform of both sides to get: s2 + 1.5=0 Solving for s I get s=i*sqrt(1.5) plugging this into the characteristic equation I get: s(t)=C1*e0*cos(sqrt(1.5)*t)+C2*e0*sin(sqrt(1.5)*t) at t=0 this equation becomes: 0=C1*e0*cos(0)+C2*e0*sin(0) therefore C1=0 so s(t)=C2*sin(sqrt(1.5)*t) Take the derivative to get: v(t)=sqrt(1.5)*C2*cos(sqrt(1.5)*t) at t=0 v(t)=20 therefore 20=sqrt(1.5)*C2*1 C2=16.33 that makes s(t)=16.33*sin(sqrt(1.5*t)) and v(t)=20*cos(sqrt(1.5)*t) take the derivative to get a(t): a(t)=-24.5*sin(sqrt(1.5)*t) All of this seemed ok to me until I graphed all three functions and realized that according to these equations when my particle is accelerating its velocity is slowing down, which can not be possible. Did I do this question completely wrong?