I am building a plane and I have a take-off distance of 50ft, is there a way for me to estimate the thrust-to-weight ratio required for this constraint (even if it was in terms of say wing loading)?
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Assuming you know the mass $m$ of the aircraft, the speed $v_0$ required to take off and the lift and drag associated.
Using basic dynamic and newton's second law you can calculate the average acceleration $a$ required to reach take off speed before the end of your runway:
$$a=\frac{v_0^2}{100\text{ft}}$$
Then using your drag generated at takeoff to be on the safe side
$$\text{Thrust} = m * a + \text{Drag}$$
Implying,
$$\frac{\text{Thrust}}{m}=a+\frac{\text{Drag}}{m}$$
