I think that the problem is that you already defined the ODE system as an anonymous function, therefore there's no need to call it using a function handle in ode45. See below.
%Variables
tspan1 = [0 0.2];
tspan2 = [0.2 0.6];
tspan3 = [0.6 1.0];
%maybe we only need three of these to just change per segment?
y0_1 = 0; %y0 (initial boundry condition of first piece)
y0_2 = 0; %y0 (initial boundry condition of second piece)
y0_3 = 0; %y0 (initial boundry condition of third piece)
ydot0_1 = 0; %This boundry condition doesn't change between pieces.
%ydot0_2 = 0;
%ydot0_3 = 0;
ics1 = [y0_1, ydot0_1]; %The intial conditions. (y(0) = ..., y'(0) = ...
%Piecewise Segments
odefcn1 = @(t,y) [y(2); -(1500.*y(1) - 1500.*(2.5.*t))./5]; %first piece
odefcn2 = @(t,y) [y(2); -(1500.*y(1) - 1500.*(0.75 - 1.25.*t))./5]; %second piece
odefcn3 = @(t,y) [y(2); -(1500.*y(1) - 1500.*(0.*t))./5]; %third piece
%ODE45 Generated Function
[t1,y1] = ode45(odefcn1, tspan1, ics1);
ics2 = y1(end,:);
ics2(1) = y0_2;
[t2,y2] = ode45(odefcn2, tspan2, ics2);
ics3 = y2(end,:);
ics3(1) = y0_3;
[t3,y3] = ode45(odefcn3, tspan3, ics3);
%Concatinate Data Values, and Plot
t = [t1;t2;t3];
y = [y1;y2;y3];
plot(t,y);
