Answer
The kinetic energy at $x = 3.0~m$ is $~~12.0~J$
Work Step by Step
To find the work done by the force from $x = 0$ to $x = 3.0~m$, we can calculate the area under the force versus position curve.
This area can be divided into three parts, including a triangle (0 to 1.0 m), a triangle (1.0 m to 2.0 m), and a rectangle (2.0 m to 3.0 m).
We can find each area separately:
$A_1 = \frac{1}{2}(4.0~N)(1.0~m) = 2.0~J$
$A_2 = \frac{1}{2}(-4.0~N)(1.0~m) = -2.0~J$
$A_3 = (-4.0~N)(1.0~m) = -4.0~J$
We can find the work done by the force:
$W = 2.0~J-2.0~J-4.0~J = -4.0~J$
The work done by the force is $~~-4.0~J$
We can find the initial kinetic energy at $x = 0$:
$K_i = \frac{1}{2}mv^2$
$K_i = \frac{1}{2}(2.0~kg)(4.0~m/s)^2$
$K_i = 16.0~J$
We can find the kinetic energy at $x = 3.0~m$:
$K_f-K_i = W$
$K_f = K_i+W$
$K_f = 16.0~J-4.0~J$
$K_f = 12.0~J$
The kinetic energy at $x = 3.0~m$ is $~~12.0~J$
