Prove the equation a. V² = u² + 2as b. s = ut + 1/2 at² 2) Derived an equation T ∝ m²lg

The equation is proven.

The equation is proven.

2) Derive an equation for the period $T$ using dimensional analysis.

The question asks to "Derive an equation $T \propto m^2lg$". However, this expression is dimensionally inconsistent for a period. Assuming the intent is to derive the formula for the period of a simple pendulum using dimensional analysis, where the period $T$ depends on mass $m$, length $l$, and acceleration due to gravity $g$, the derivation is as follows:

Let $T = k m^a l^b g^c$, where $k$ is a dimensionless constant and $a, b, c$ are exponents to be determined.
The dimensions of the quantities are:

For M: $a = 0$
For L: $b+c = 0$
For T: $1 = -2c$

From the T equation, $c = -\frac{1}{2}$.
Substitute $c$ into the L equation: $b + (-\frac{1}{2}) = 0 \implies b = \frac{1}{2}$.
So, the exponents are $a=0$, $b=\frac{1}{2}$, $c=-\frac{1}{2}$.

The derived equation for the period of a simple pendulum is $\boxed{T = k \sqrt{\frac{l}{g}}}$, where $k$ is a dimensionless constant (experimentally found to be $2\pi$).