A foggy game of chicken

Classical Mechanics Level 5

You're the captain of a shipping boat making your approach to harbor, and you're close to your delivery deadline. To finish the delivery, you have to sail underneath a bridge that closes every so often in order to let ships through in the other direction. Exactly $$t^*$$ seconds before the bridge closes, a light turns from green to yellow to indicate ships to use caution. Because of the heavy fog, you can't see the light until you're a distance $$d$$ from the bridge, at which point you determine it is yellow.

If you make it through the bridge before it closes (in which case your delivery will be early) you receive a bonus of $100,000. If the light turns red (indicating the passage is closed) before you pass through, you must bring the ship to a complete stop (to avoid crashing). If you do this, you shed all of the kinetic energy your ship had, and you'll have to spend fuel to bring it back up to the original cruising speed. What cruising speed, $$v$$ (in m/s), should your ship maintain to maximize the expected value of passing under the bridge? Details • Your ship weighs $$M=10^8$$ kg • Your on-time bonus is$100,000
• For simplicity, assume that the price of fuel is $$\gamma_E = \0.1$$/kJ
• $$d = 1500$$ m
• $$t^* = 1000$$ s
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