  Van der Waals interactions with transition metal dichalcogenides was shown to
induce strong spin-orbit coupling (SOC) in graphene, offering great promises to
combine large experimental flexibility of graphene with unique tuning
capabilities of the SOC that can rotate spin by moving electrons or vice versa.
Here, we probe SOC-driven band splitting and electron dynamics in graphene on
WSe$_2$ by measuring ballistic transverse magnetic focusing. We found a clear
splitting in the first focusing peak whose evolution in charge density and
magnetic field is well reproduced by calculations using SOC strength of ~13 meV
and no splitting in the second peak that indicates stronger Rashba SOC. A
possible suppression of electron-electron scatterings was also found in
temperature dependence measurement. Further, we found that Shubnikov-de Haas
oscillations exhibit SOC strength of ~3.4 meV, suggesting that it probes
different electron dynamics, calling for new theory. Our study demonstrates an
interesting possibility to exploit ballistic electron motion pronounced in
graphene for emerging spin-orbitronics.
