Which of the following is dimensionless?

Options

A. Force/acceleration
B. Velocity/acceleration
C. Volume/area
D. Energy/work

Answer

To determine which of the given options is dimensionless, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Force per unit acceleration**:
- Force (F) has the dimension of mass times acceleration:
\[ [F] = [M][L][T^{-2}] \]
- Acceleration (a) has the dimension:
\[ [a] = [L][T^{-2}] \]
- Therefore, force per unit acceleration is:
\[ \frac{[F]}{[a]} = \frac{[M][L][T^{-2}]}{[L][T^{-2}]} = [M] \]
- This is not dimensionless.

2. **Velocity per unit acceleration**:
- Velocity (v) has the dimension:
\[ [v] = [L][T^{-1}] \]
- Using the same dimension for acceleration as before:
\[ [a] = [L][T^{-2}] \]
- Therefore, velocity per unit acceleration is:
\[ \frac{[v]}{[a]} = \frac{[L][T^{-1}]}{[L][T^{-2}]} = [T] \]
- This is not dimensionless.

3. **Volume divided by area**:
- Volume (V) has the dimension:
\[ [V] = [L^3] \]
- Area (A) has the dimension:
\[ [A] = [L^2] \]
- Therefore, volume divided by area is:
\[ \frac{[V]}{[A]} = \frac{[L^3]}{[L^2]} = [L] \]
- This is not dimensionless.

4. **Energy divided by work**:
- Energy (E) and work (W) are equivalent in terms of dimensions:
\[ [E] = [W] = [M][L^2][T^{-2}] \]
- Therefore, energy divided by work is:
\[ \frac{[E]}{[W]} = \frac{[M][L^2][T^{-2}]}{[M][L^2][T^{-2}]} = 1 \]
- This means it is dimensionless.

### Conclusion:
The only option that is dimensionless is **Energy divided by work**.