Calculate constant acceleration needed to reach terminal velocity at specific time [closed]

Question

I'm trying to find a way to calculate a constant acceleration given the following:

One dimensional motion.

The only forces involved are the force ($F_a$) caused by a constant acceleration ($a_c$) and the drag ($F_d$) against it.

$a_c$ - Constant acceleration

$v_o$ - Initial velocity

$v_t$ - Terminal velocity

$m$ - Mass of object

$\rho$ - Density of fluid

$A$ - Projected area of the object

$C_d$ - Drag coefficient

$t_t$ - Time to reach n% (e.g. 95%) of terminal velocity

$v_o = 0\,m/s$

$v_t = 300\,m/s$

$m = 1000\, kg$

$\rho = 1\, kg/m^3$

$A = 100\, m^2$

$C_d = 0.5$

$t_t = 4\, sec$

Terminal Velocity: $$ v_t = \sqrt{2 m a_c \over \rho A C_d} $$

Drag Force: $$ F_d = {1\over2}\rho A C_d v^2 $$

Solving for $a_c$ using the terminal velocity equation is easy enough ($ a_c = { \rho A C_d v_t^2 \over 2 m } = 2250\,m/s^2$), but that doesn't correctly account for the target time ($t_t$).

How can I find the correct $a_c$ to reach n% of $v_t$ at time $t_t$? I'm assuming $m$, $A$, and/or $C_d$ will need to be adjusted to achieve this, but I'm having trouble figuring out how to bring the target time concept into the mix in the first place.

Update: Related question: Calculating time to reach certain velocity with drag force, but 1) I'm having trouble fully grasping the math that's going on there. 2) It's solving for the time required to reach a specific velocity, whereas I'm trying to find the necessary constant acceleration to reach a specific velocity at a specific time.

Answer 1

Step One: Writing Equation of Motion

Step Two: Solving Equation of Motion for Velocity

Both these steps are shown in the following Mathematica link where the Equation is typed in the upper portion of the link and the functional form of the velocity as a function of time is calculated:

Equation of Motion and Its Solution

Known Quantities Plugged In

Using Initial Value Condition v(0)=0

V(t) = V_Terminal * Tanh(0.158114*Sqrt(a)*t)

,where V_terminal = 6.32456*Sqrt(a) Since taking the limit of velocity for very large time scales would identically be equal to unity.

Now you can use these last two equations knowing n% of V_terminal obtained after t_t seconds.