You're super close...
Just make one change to the height function so it looks like this one below:
function [h] = height(a,t,h0,v0,t0)
(Notice the .^ instead of ^ to signify element-wise exponentiation. This way you can pass entire vectors in place of t.)
Then, modify the main code by remove the for-loop and instead use something like
F=input('Engine force (N): ');
m=input('Mass (kg): ');
dt=input('Precision/timesteps (s): ');
t = t0: dt: 0.15;
Notice that t is now a vector and that I'm only calling the plot function once, the first curve (height vs. t) is blue ('b') and the second curve (velcoity1 vs t) is red. Hopefully that helps.