Question

1. If momentum P, Area A and time T are taken to be fundamental quantities, then energy
has the dimensional formula-​

Answer 1

Well, kinetic energy is a form of energy.

$E=\dfrac {1}{2}mv^2$

Or,

$E=\dfrac {1}{2}pv\quad\quad [\because\ p=mv]$

But we know that,

$\mathrm {Velocity=\dfrac {displacement\ (length)}{time}}\\\\\\v=\dfrac {l}{t}$

Then,

$E=\dfrac {1}{2}\cdot p\cdot\dfrac {l}{t}$

Well, $l^2$ is similar to area, $A.$ So, can I say that $\sqrt{A}=A^{\frac {1}{2}}$ is similar to $l$ ?!

Thus,

$E=\dfrac {1}{2}\cdot p\cdot\dfrac {A^{\frac {1}{2}}}{t}$

Finally, the dimension for energy,

$\large\boxed {\mathbf {[E]=\left[P\ A^{\frac {1}{2}}\ T^{-1}\right]}}$