Question

Proof that v=√T/√M

Where, v=velocity, T=tension and M=mass

Answer 1

Rewrite the equation according to math rules:

$$v=\frac{\sqrt T}{\sqrt \Mu}=\sqrt{\frac{T}{\Mu}}.$$

Choose unusual approach. We know that tension is measures in N (newtons), mass per unit length is kg/m (kilograms per metre).

What is a newton?

$$[N]=\frac{[kg]·[m]}{[s]^2}.$$

Divide this by kg/m:

$$\frac{T}{\Mu}=\frac{[kg]·[m]}{[s]^2}·\frac{[m]}{[kg]}=\frac{[m]^2}{[s]^2}.$$

Now take a square root of this:

$$\sqrt{\frac{[m]^2}{[s]^2}}=\text{m/s}.$$

We see that the velocity is square root of tension divided my linear mass, which is m/s. We know that velocity is measured in m/s, so, we proved that

$$v=\frac{\sqrt T}{\sqrt \Mu}$$