We explore a natural interplay between the solution to the time-dependent free Schrodinger equation and the (spatial) X-ray transform– proving a variant on a Strichartz estimate. Our estimates are sharp in the sense that we identify the best constant C and show that an initial condition f achieves equality in the estimate if and only if it is an isotropic centered Gaussian. In higher dimensions, we prove similar results where the X-ray transform is replaced by the more general k-plane transform. In the process, we obtain sharp L^2(\mu) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures supported on natural ``co-k -planarity” sets. This is joint work with Jonathan Bennett, Neal Bez, Susana Gutierrez, and Marina Iliopoulou.