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We study the capacity of ad hoc wireless networks with mobile nodes. The mobility model examined is one where the nodes are restricted to move along one-dimensional paths. We examine the scaling laws for the per user throughput achievable over long time-scales, making this suitable for applications with loose delay constraints. We show that under this regime of restricted mobility, we attain a constant throughput (i.e., Θ (1)) per user, which is significantly higher than the throughput of fixed networks, which decays as O(1/√n) with the number of nodes n, as shown by Gupta and Kumar.