Delta t = 1 second

Clear["Global`*"]

Initial conditions in MKS units

m = 70;
c = 0.3;
g = 9.8;

dt = 1;

Initialize arrays

t = x = v = a = Table[null,{i,0,10/dt}];

t[[0]] = 0;
x[[0]] = 0;
v[[0]] = 0;

Do Loop

Do[

a[[i-1]] = -(c v[[i-1]]^2)/m + g;
t[[i]] = t[[i-1]] + dt;
x[[i]] = x[[i-1]] + v[[i-1]] dt;
v[[i]] = v[[i-1]] + a[[i-1]] dt;

,{i,1,10/dt}]

table1 = Table[{i,x[[i/dt]]},{i,1,10}];
table1//TableForm

Delta t = 0.1 second

Clear[t,x,v,a]

Initial conditions in MKS units

m = 70;
c = 0.3;
g = 9.8;

dt = 0.1;

Initialize arrays

t = x = v = a = Table[null,{i,0,10/dt}];

t[[0]] = 0;
x[[0]] = 0;
v[[0]] = 0;

Do Loop

Do[

a[[i-1]] = -(c v[[i-1]]^2)/m + g;
t[[i]] = t[[i-1]] + dt;
x[[i]] = x[[i-1]] + v[[i-1]] dt;
v[[i]] = v[[i-1]] + a[[i-1]] dt;

,{i,1,10/dt}]

table2 = Table[{i,x[[i/dt]]},{i,1,10}];
table2//TableForm

Delta t = 0.01 second

Clear[t,x,v,a]

Initial conditions in MKS units

m = 70;
c = 0.3;
g = 9.8;

dt = 0.01;

Initialize arrays

t = x = v = a = Table[null,{i,0,10/dt}];

t[[0]] = 0;
x[[0]] = 0;
v[[0]] = 0;

Do Loop

Do[

a[[i-1]] = -(c v[[i-1]]^2)/m + g;
t[[i]] = t[[i-1]] + dt;
x[[i]] = x[[i-1]] + v[[i-1]] dt;
v[[i]] = v[[i-1]] + a[[i-1]] dt;

,{i,1,10/dt}]

table3 = Table[{i,x[[i/dt]]},{i,1,10}];
table3//TableForm

True Theory

Clear[t,x,v,a]

x[t_] = Sqrt[m g/c] (t + Sqrt[m g/c]/g *
  Log[(1 + Exp[-2 g t/Sqrt[m g/c]])/2])

table4 = Table[{t,x[t]},{t,1,10}];
table4//TableForm

All the tables at once

Table[{i,table1[[i,2]],table2[[i,2]],table3[[i,2]],
         table4[[i,2]]},{i,1,10}]//TableForm

Plots

plot1 = ListPlot[table1,PlotJoined->True];

plot2 = ListPlot[table2,PlotJoined->True];

plot3 = ListPlot[table3,PlotJoined->True];

plot4 = ListPlot[table4,PlotJoined->True];

Show[plot1,plot2,plot3,plot4];