Velocity Augmentation Suit

There are two equations, one for velocity on x-axis and other for velocity on y-axis (note: $\dot{x}$ means the time derivative of $x)$ :

$\large \displaystyle \dot{v_x} = \frac{\alpha}{m}v_x$

$\large \displaystyle \dot{v_y} = \frac{\alpha}{m}v_y - g$

First one leads to:

$\large \displaystyle v_x(t) = v_{0x} e^{(\alpha/m) t}$

$\color{#20A900} \boxed{\large \displaystyle v_x(t) = 10 e^{(\alpha/150) t}}$

While second leads to:

$\large \displaystyle \frac{dv_y}{v_y - gm/ \alpha} = (\alpha /m) dt$

Integrating both sides:

$\large \displaystyle \ln \left (v_y - gm/ \alpha \right ) \Bigg |_{20}^{v_y} = \alpha t/m$

$\large \displaystyle v_y(t) = gm/ \alpha + (20 - gm/ \alpha) e^{(\alpha/m) t}$

$\color{#20A900} \boxed{\large \displaystyle v_y(t) = (1500/ \alpha) + (20 - 1500/ \alpha) e^{(\alpha /150)t}}$

The angle with respect to the horizontal has its tangent equal to:

$\large \displaystyle \tan[\theta(t)] = \frac{v_y(t)}{v_x(t)}$

$\color{#20A900} \boxed{\large \displaystyle \tan[\theta(t)] = (1500/ \alpha) e^{-(\alpha /150)t} + (2-150/ \alpha)}$

We want the limit of the angle when time approaches infinity to be $0$ . So:

$\large \displaystyle \lim_{t \rightarrow \infty} \tan[\theta(t)] = 2 - 150/ \alpha = 0$

Thus:

$\color{#3D99F6} \boxed{\large \displaystyle \alpha = 75}$