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].
