Understanding Kinetic Energy
Kinetic energy (KE) is defined as the energy possessed by an object due to its motion. The formula for kinetic energy is given by:
\[ KE = \frac{1}{2}mv^2 \]
where \( m \) is mass and \( v \) is velocity.

Identifying Fundamental Quantities
In this scenario, we will consider velocity \( p \), acceleration \( q \), and density \( r \) as the fundamental quantities.

Dimensional Analysis
1. **Velocity (p)**:
- Dimensional formula: \( [L^1 T^{-1}] \)
2. **Acceleration (q)**:
- Dimensional formula: \( [L^1 T^{-2}] \)
3. **Density (r)**:
- Dimensional formula: \( [M^1 L^{-3}] \)

Expressing Kinetic Energy in Terms of p, q, r
To derive the dimensional formula for kinetic energy in terms of \( p \), \( q \), and \( r \):
- **Mass (m)** can be derived from density:
\[ m = V \cdot r = [L^3] \cdot [M^1 L^{-3}] = [M^1] \]
Thus, \( m \) can be expressed in terms of \( r \) and volume.
- **Kinetic Energy (KE)**:
\[ KE \sim m v^2 = r(p^2) \]
To express mass in terms of \( r \) and volume, we can use \( r = \frac{m}{V} \) and rearrange.
- **Final Representation**:
The dimensional formula for kinetic energy becomes:
\[ KE \sim p^8 r q^{-3} \]

Conclusion
The dimensional formula for kinetic energy, expressed in terms of velocity \( p \), acceleration \( q \), and density \( r \), is indeed \( p^8 r q^{-3} \). This encapsulates the relationship between these fundamental quantities and kinetic energy effectively.