Final answer:
The position with the greatest value of x at which the particle comes to rest is x = 4.0 m.
Step-by-step explanation:
To determine the position where the particle comes to rest, we need to find the point where the particle's velocity becomes zero. The potential energy function provided, u(x), can be differentiated with respect to x to obtain the force function, F(x). The force function can then be used to find the acceleration, a(x), of the particle. Since the particle is initially moving with a speed of 30.0 cm/s, it will eventually come to rest at a position where the acceleration is zero. This occurs when the force exerted on the particle is zero. So, we can set F(x) equal to zero and solve for x to find the position where the particle comes to rest.
F(x) = -dU(x)/dx
a(x) = F(x)/m = 0
In this case, the potential energy function is given by u(x) = -(2.0 j·m)/x + (4.0 j·m^2)/x^2. Differentiating u(x) with respect to x and setting it equal to zero, we find:
dU(x)/dx = -(2.0 j·m)/x^2 + (8.0 j·m^2)/x^3 = 0
Multiplying through by x^3 to eliminate the denominators, we get:
-2.0 j·m + 8.0 j·m^2/x = 0
Combining like terms and rearranging, we get:
8.0 j·m^2 = 2.0 j·m
Dividing through by 2.0 j·m, we find:
4.0 = x
Therefore, the position with the greatest value of x at which the particle comes to rest is x = 4.0 m.