The displacement of a particle is given by `x=At^(3)+Bt^(2)+Ct+D`. The dimension of L in `(AD)/(BC)` will be

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the problem, we need to analyze the given displacement equation and the expression \((AD)/(BC)\) to find the dimensions of length \(L\).

### Step 1: Identify the dimensions of each term in the displacement equation

The displacement \(x\) is given by:
\[ x = At^3 + Bt^2 + Ct + D \]

Here, \(A\), \(B\), \(C\), and \(D\) are constants, and \(t\) is time.

1. **Displacement \(x\)** has the dimension of length \([L]\).
2. **Time \(t\)** has the dimension of time \([T]\).

Now, we will find the dimensions of \(A\), \(B\), \(C\), and \(D\) based on the terms they multiply.

- For the term \(At^3\):
\[ [A][T]^3 = [L] \implies [A] = \frac{[L]}{[T]^3} \]

- For the term \(Bt^2\):
\[ [B][T]^2 = [L] \implies [B] = \frac{[L]}{[T]^2} \]

- For the term \(Ct\):
\[ [C][T] = [L] \implies [C] = \frac{[L]}{[T]} \]

- The constant \(D\) is also a displacement, so:
\[ [D] = [L] \]

### Step 2: Substitute the dimensions into the expression \((AD)/(BC)\)

Now we will substitute the dimensions we found into the expression \((AD)/(BC)\):

1. **Calculate the dimensions of \(AD\)**:
\[ [AD] = [A][D] = \left(\frac{[L]}{[T]^3}\right)[L] = \frac{[L]^2}{[T]^3} \]

2. **Calculate the dimensions of \(BC\)**:
\[ [BC] = [B][C] = \left(\frac{[L]}{[T]^2}\right)\left(\frac{[L]}{[T]}\right) = \frac{[L]^2}{[T]^3} \]

### Step 3: Find the dimensions of \((AD)/(BC)\)

Now we can find the dimensions of the expression:
\[ \frac{[AD]}{[BC]} = \frac{\frac{[L]^2}{[T]^3}}{\frac{[L]^2}{[T]^3}} = 1 \]

### Conclusion

The dimension of \((AD)/(BC)\) is dimensionless (1).