Final answer:
To find the speed of the body at x = 2.0 m, we need to find the force acting on the body at x = 2.0 m and then use the equations of motion to calculate the speed. The force acting on the body at x = 2.0 m is -8.0 N, and the acceleration is -8.0 m/s². Using the equation v² = u² + 2as, we can find the speed at x = 2.0 m to be 2sqrt(10) m/s.
Step-by-step explanation:
To find the speed of the body at x = 2.0 m, we first need to find the force acting on the body at x = 2.0 m using the given force equation: F(x) = -4.0x. Substituting x = 2.0 into the equation, we get F(2.0) = -4.0(2.0) = -8.0 N.
Next, we can use the equation F = ma, where F is the force, m is the mass, and a is the acceleration, to find the acceleration of the body. Since m = 1.0 kg, we can rearrange the equation to a = F/m and substitute the values to get a = -8.0/1 = -8.0 m/s².
Finally, we can use the equation v² = u² + 2as to find the speed at x = 2.0 m, where u is the initial velocity (given as 4.0 m/s) and s is the displacement. Since the body starts at x = 3.5 m and moves to x = 2.0 m, the displacement is -1.5 m. Substituting the values, we have v² = 4.0² + 2(-8.0)(-1.5) = 16 + 24 = 40. Taking the square root of both sides, we get v = sqrt(40) = 2sqrt(10) m/s.