How did Rutherford in the Gold Foil experiment change the input Alpha particle energy without particle accelerators? If they did, how would they even know the speeds? . This graph shows scattering of high energy alpha particles at a fixed angle which can be used to estimate the radius of the nucleus. This was possible performed during the gold foil scattering experiment. How/did they change the input energy if it was just alpha particles decaying from some radium.
In order to determine the variation of scattering with velocity the apparatus was somewhat modified. A conical glass tube coated with active deposit was again used as source of radiation. This source was placed about 1 mm. from the mica window (E, fig. 3), so that it was possible to insert additional sheets of mica between the source and the window to reduce the velocity of the α particles. Mica sheets were used for this purpose on account of their uniformity of thickness in comparison with metal foils.
The data in that plot decidedly do not come from a Geiger-Marsden style apparatus (which is limited to a little over 5 MeV by the available alpha decay sources).
The plotted data was likely taken sometime after dual van~de~Graff accelerators became available. In short the experiment in Rutherford's lab didn't find the nuclear radius: they only put a upper limit on it that was much smaller than the known size of the atoms.
It is always possible to control the speed of charged particles, like the alpha, with an accelerating grid in the same fashion as in the Franck-Hertz experiment. To my knowledge, the original Geiger-Marsden-Rutherford experiments, unlike the data you present, focused on the angular dependency of the cross section at constant initial alpha velocity [see wiki Geiger–Marsden experiment].