The motion of a particle of mass m is given by `x=0` for `tlt0s`,`x(t)=Asin4pit` for `0lttlt((1)/(4))s `
`(Agt0)` and `x=0` for `tgt((1)/(4))s`.

Options

A. The force at `t=((1)/(8))`s on the particle is `-16pi^2A-m`
B. The particle is acted upon by on impulse of magnitude `4pi^2A-m` at `t=0s` and `t=((1)/(4))`s
C. The particle is not acted upon by any force
D. The particle is not acted upon by a constant force

Explanation

To solve the problem step by step, we will analyze the motion of the particle given by the equations provided.

### Step 1: Understand the motion of the particle
The motion of the particle is defined in three intervals:
1. For \( t < 0 \): \( x = 0 \)
2. For \( 0 < t < \frac{1}{4} \): \( x(t) = A \sin(4 \pi t) \) (where \( A > 0 \))
3. For \( t > \frac{1}{4} \): \( x = 0 \)

### Step 2: Find the velocity of the particle
To find the velocity \( v(t) \), we differentiate the position function \( x(t) \) with respect to time \( t \):
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(A \sin(4 \pi t)) = 4 \pi A \cos(4 \pi t) \]

### Step 3: Find the acceleration of the particle
Next, we differentiate the velocity function \( v(t) \) to find the acceleration \( a(t) \):
\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(4 \pi A \cos(4 \pi t)) = -16 \pi^2 A \sin(4 \pi t) \]

### Step 4: Evaluate acceleration at \( t = \frac{1}{8} \) seconds
Now, we will evaluate the acceleration at \( t = \frac{1}{8} \) seconds:
\[ a\left(\frac{1}{8}\right) = -16 \pi^2 A \sin\left(4 \pi \cdot \frac{1}{8}\right) = -16 \pi^2 A \sin\left(\frac{\pi}{2}\right) = -16 \pi^2 A \]

### Step 5: Calculate the force acting on the particle
Using Newton's second law, we can find the force \( F \) acting on the particle:
\[ F = m \cdot a = m \cdot (-16 \pi^2 A) = -16 \pi^2 m A \]

### Step 6: Analyze the options
Now we can analyze the options given in the problem based on our calculations:
- **Option A**: The force at \( t = \frac{1}{8} \) seconds is \( -16 \pi^2 m A \) (Correct)
- **Option B**: The impulse at \( t = 0 \) and \( t = \frac{1}{4} \) is the same (Correct, since the impulse depends on the change in momentum)
- **Option C**: The particle is acted upon by a constant force (Incorrect, since the force changes with time)
- **Option D**: The particle is not acted upon by any constant force (Correct)

### Conclusion
The correct options are A, B, and D.