The SI unit of energy is `J = kg m^2 s^(-2),` that of speed `upsilon` is `ms^(-1)` and of acceleration a is `ms^(-2)` which of the formulae for kinetic energy (K) given below can you rule out on the basis of dimensional arguments (m stands for the mass of the body).
(a) `K = m^2 upsilon^3` (b) `K = (1)/(2)m upsilon^2` (c ) K= ma
(d) `K =(3)/(16)m upsilon^2` (e ) `K = (1)/(2) m upsilon^2 +ma`

Explanation

To determine which of the provided formulas for kinetic energy can be ruled out based on dimensional analysis, we need to ensure that the dimensions on both sides of each equation match. The SI unit of energy (kinetic energy) is Joules (J), which has the dimensional formula of \( [K] = [M][L^2][T^{-2}] \), or in terms of base units, \( kg \, m^2 \, s^{-2} \).

We will analyze each option step by step:

### Step-by-Step Solution:

1. **Option (a): \( K = m^2 \upsilon^3 \)**
- Dimensions:
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