projectile question (calculate return time and compare them)

I Have a question from a textbook

where some equations are written wrong. The correct versions of them are as follows:

v(t)=(-mg/k)+(v0+(mg/k))*(1-exp(-kt/m))

y(t)=-(m-g-t/k)+(m/k)(v0+(mg/k))(1-exp(-kt/m))

for part-c

v_bar = - gt +v0

y_bar = - ((gt^2)/2) + v0t

I do the first and second parts (a, b) as follows. But I am not definitely sure about my results.

In addition to this, I could not create any code for part-c. Please I'm asking for a help to do such type question.

function projectile  = max_height( m, k )
%UNTITLED Summary of this function goes here
%   Detailed explanation goes here


g = 9.8;
v0 = 25;
t = 0:0.1:12;

 v = - (m*g/k) + ((v0 + (m * g/k)) * exp(- k * t /m));
y = - (m*g*t/k) +((m/k)*(v0 +(m*g/k))*(1-exp(- k * t /m)));
 

plot(t,y);

[max_height, t] = max(y)

end

[darova]

Do you know how to calculate return time?

[Zeynep Toprak]

no I dont know in fact. I'm just learner of matlab. I have no idea about physics.

[darova]

This is it? I highlighted in red

[Zeynep Toprak]

well, perfect! I see. But, how can I find it with formula?

[darova]

How to find second point?

[Zeynep Toprak]

v(t)=0. Then I will find the root of v(t)=0 for t. That's, I will draw t from v(t)=0. Right?

[darova]

I believe that you should find it numerically. But velocity will never be 0

MATLAB doesn't know where is the ground

You can use find to find first negative value

plot(t,y);
ix = find(y < 0,1);
hold on
plot(t(ix),y(ix),'or')
hold off