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We investigate the energy-time efficiency of two reactive route discovery protocols (BERS and DTC) with chase packets in the expanding ring search (ERS). BERS introduces deliberately waiting time in broadcasts of RREQs. DTC uses a threshold to switch between two different channel speeds to allow the chase packets to catch up with fulfilled RREQs. Taking an analytical approach, we gain the insight into these protocols and explore the trade-off nature of the energy and time efficiency. An optimal threshold for DTC is identified, and the performance of DTC under this optimal condition is used to compare with the performance of BERS in the light of ERS. Our results have practical values and implications as well as theoretical interests. We found that BERS has achieved the best overall performance in terms of energy-time efficiency, while DTC achieved the least overall performance despite the reduction of latency.