You made the classic mistake of thinking that this set of logical statements would be handled for each element of your vector. But Matlab doesn't work that way, unless you tell it with a for-loop.
The code below uses logical indexing to get the result.
t=-5:50;
v=zeros(size(t));
L=t<=8;
v(L)=10.*(t(L).^2)-(5.*t(L));
L=(t>=8) & (t<=16);
v(L)=624-(5.*t(L));
L=(t>=16) & (t<=26);
v(L)=(36.*t(L))+12.*(t(L)-16).^2;
L=t>26;
v(L)=2136.*exp(-0.1.*(t(L)-26));
L=t<0;
v(L)=0;
plot(t,v)
