Block for plotting a function using different parameters

Question

I want to plot height[time] function for different values of M and g. This function contains a[2] which needs to be found. I had two-coupled algebraic equation for a[2] and t[2]. I used Solve to solve that system and it worked. However the issue starts at that point, when I execute below code, it does not produce a graph. I did something wrong, but the question is "where". Please see below;

p01[x_] := Integrate[2*a[2]/k*1/(x - c) + t[2]/k , {x, 0, x},Assumptions -> 0 < x < c] + 1/2

p02[x_] := Integrate[2*a[2]/k*1/(x - c) + t[2]/k , {x, 1, x},Assumptions -> c < x < 1] 

first[x_] := Integrate[p01[x], {x, 0, c}, Assumptions -> 0 < c < 1]

second[x_] := Integrate[p02[x], {x, c, 1}, Assumptions -> 0 < c < 1]

third[x_] := Integrate[(x - c)*p01[x], {x, 0, c}, Assumptions -> 0 < c < 1]

fourth[x_] := Integrate[(x - c)*p02[x], {x, c, 1}, Assumptions -> 0 < c < 1]

Solve[2*M*a[2] == first[x]+second[x] - M*g && 2*M*i*t[2] ==third[x]+fourth[x], {a[2], t[2]}]

c = 0.4; k = 1; i = 0.2;

a[0] = k*c*(1 - c);
t[0] = k*(1 - 2*c);

height[time_] := a[0] + time^2*a[2]

plot4 = Block[{M = {0, 0.1, 1, 10}, g = {0, 0.1, 1, 10}}, 
  Plot[Evaluate@height[time], {time, 0, 1}, AxesOrigin -> {0, 0}, 
   PlotStyle -> {Red, Green, Blue, Black}]]

Answer 1

Look, there are several issues with your code which is why it is important that you tear the pieces apart before posting a question. I'm sure you could have solved this yourself.

1. You don't use the solution for a[2] and t[2]. You call Solve correctly, but you don't store the solution and use it in Plot.
2. You have a mystical b in your code-block that never gets a value.

Want to know how you easily find all this issues? One easy way is always to not use Plot because then you only see an empty graph. What I usually do is to type Plott which is a function that is not defined and therefore, nothing happens except that you really see the parameters. So after evaluating your code-block do (see the Plott!):

Now you see that there is an unevaluated a[2] in there. So the first thing you do is to store the solution of Solve

sol = 
 Flatten@Solve[
   2*M*a[2] == first[x] + second[x] - M*g && 
    2*M*i*t[2] == third[x] + fourth[x], {a[2], t[2]}]

and use it

Now, you see that your formula contains the b which needs a value. I'll set it to b=1 in the Block and replace Plott with Plot again and there you have it