A particle is moving with the given acceleration function and initial conditions. Find the velocity function and the position function of this particle. a(t) = 2 sin(t) − 4 cos(t), v(0) = −3, s(0) = 5

Answer:The velocity function is the particle is [tex]v(t)=-2\cos (t)-4\sin (t)-1[/tex].Step-by-step explanation:The acceleration function of a moving particle is[tex]a(t)=2\sin (t)-4\cos (t)[/tex]The initial conditions are v(0) = −3, s(0) = 5.Integrate the acceleration function with respect to time to find the velocity function.[tex]\int a(t)=\int (2\sin (t)-4\cos (t))dt[/tex][tex]v(t)=-2\cos (t)-4\sin (t)+C_1[/tex]Use the initial condition v(0) = −3 to find the value of C₁.[tex]-3=-2\cos (0)-4\sin (0)+C_1[/tex][tex]-3=-2(1)-4(0)+C_1[/tex][tex]-3=-2+C_1[/tex][tex]-3+2=C_1[/tex][tex]-1=C_1[/tex]So the velocity function is the particle is[tex]v(t)=-2\cos (t)-4\sin (t)-1[/tex]Integrate the acceleration function with respect to time to find the position function.[tex]\int v(t)=\int (-2\cos (t)-4\sin (t)-1)dt[/tex][tex]s(t)=-2\sin (t)+4\cos (t)-t+C_2[/tex]Use the initial condition s(0) = 5 to find the value of C₂.[tex]5=-2\sin (0)+4\cos (0)-(0)+C_2[/tex][tex]5=-2(0)+4(1)+C_2[/tex][tex]5=4+C_2[/tex][tex]1=C_2[/tex]So, the position function is the particle is [tex]s(t)=-2\sin (t)+4\cos (t)-t+1[/tex].