Question

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 1

Answer:

The velocity function is the particle is $v(t)=-2\cos (t)-4\sin (t)-1$.

Step-by-step explanation:

The acceleration function of a moving particle is

$a(t)=2\sin (t)-4\cos (t)$

The initial conditions are v(0) = −3, s(0) = 5.

Integrate the acceleration function with respect to time to find the velocity function.

$\int a(t)=\int (2\sin (t)-4\cos (t))dt$

$v(t)=-2\cos (t)-4\sin (t)+C_1$

Use the initial condition v(0) = −3 to find the value of C₁.

$-3=-2\cos (0)-4\sin (0)+C_1$

$-3=-2(1)-4(0)+C_1$

$-3=-2+C_1$

$-3+2=C_1$

$-1=C_1$

So the velocity function is the particle is

$v(t)=-2\cos (t)-4\sin (t)-1$

Integrate the acceleration function with respect to time to find the position function.

$\int v(t)=\int (-2\cos (t)-4\sin (t)-1)dt$

$s(t)=-2\sin (t)+4\cos (t)-t+C_2$

Use the initial condition s(0) = 5 to find the value of C₂.

$5=-2\sin (0)+4\cos (0)-(0)+C_2$

$5=-2(0)+4(1)+C_2$

$5=4+C_2$

$1=C_2$

So, the position function is the particle is $s(t)=-2\sin (t)+4\cos (t)-t+1$.