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We present SiO J=2-1 maps of the Sgr B2 molecular cloud, which show shocked gas with a turbulent substructure comprising at least three cavities at velocities of [10,40] km s$^{-1}$ and an arc at velocities of [-20,10] km s$^{-1}$. The spatial anti-correlation of shocked gas at low and high velocities, and the presence of bridging features in position-velocity diagrams suggest that these structures formed in a cloud-cloud collision. Some of the known compact HII regions spatially overlap with sites of strong SiO emission at velocities of [40,85] km s$^{-1}$, and are between or along the edges of SiO gas features at [100,120] km s$^{-1}$, suggesting that the stars responsible for ionizing the compact HII regions formed in compressed gas due to this collision. We find gas densities and kinetic temperatures of the order of $n_{\rm H_2}\sim 10^5\rm cm^{-3}$ and $\sim$30 K, respectively, towards three positions of Sgr B2. The average values of the SiO relative abundances, integrated line intensities, and line widths are $\sim$10$^{-9}$, $\sim$11 K km s$^{-1}$, and $\sim$31 km s$^{-1}$, respectively. These values agree with those obtained with chemical models that mimic grain sputtering by C-type shocks. A comparison of our observations with hydrodynamical simulations shows that a cloud-cloud collision that took place $\lesssim$ 0.5 Myr ago can explain the density distribution with a mean column density of $\bar{N}_{\rm H_2}\gtrsim 5\times10^{22}$ cm$^{-2}$, and the morphology and kinematics of shocked gas in different velocity channels. Colliding clouds are efficient at producing internal shocks with velocities $\sim$5-50 km $s^{-1}$. High-velocity shocks are produced during the early stages of the collision and can readily ignite star formation, while moderate- and low-velocity shocks are important over longer timescales and can explain the widespread SiO emission in Sgr B2.