The acceleration of a particle is defined by the relation a = -60x^-1.5, where a is expressed in m/s^2 and x in meters. Knowing that the particle starts with no initial velocity at x = 4 m, determine the velocity of the particle when x = 2 m, x = 1 m, x = 100 mm.

To find the velocity of the particle at different positions, we need to integrate the given acceleration function. The initial condition tells us that the particle starts with no initial velocity at x = 4 m.

To find the velocity function (v) as a function of position (x), we will integrate the acceleration function (a) with respect to x:

∫ a dx = ∫ (-60x^(-1.5)) dx

Using the power rule of integration, we get:

v(x) = -120x^(-0.5) + C

where C is the constant of integration. To find the value of C, we can use the given initial condition that the particle starts with no initial velocity at x = 4 m. Let's substitute these values into our equation:

v(4) = -120(4)^(-0.5) + C
0 = -120(2) + C
C = 240

Now that we have found the value of C, we can use it to find the velocity at different positions.

1. Velocity at x = 2 m:
v(2) = -120(2)^(-0.5) + 240
= -120 * (√2)^(-1) + 240
≈ -169.71 m/s

2. Velocity at x = 1 m:
v(1) = -120(1)^(-0.5) + 240
= -120 * (√1)^(-1) + 240
≈ 120 m/s

3. Velocity at x = 100 mm (converted to meters):
v(0.1) = -120(0.1)^(-0.5) + 240
= -120 * (√0.1)^(-1) + 240
≈ 379.67 m/s

Therefore, the velocity of the particle at x = 2 m is approximately -169.71 m/s, at x = 1 m is approximately 120 m/s, and at x = 100 mm (0.1 m) is approximately 379.67 m/s.