I was on my way to Norbury and was passing through Pollards Hill, an unremarkable estate near Mitcham, when I realised something. The 152 and 255 both start their routes by taking a large diversion around part of the estate in a horseshoe shape, before returning to the road they started on. Therefore, you could miss your bus and yet still catch it, provided you walked along the road quickly enough.
Obviously you want to catch the next bus, yet you also want to minimise the distance travelled as this makes it more likely you'll catch the bus. In the example given, the distance between the first bus stop and the next following stop on the same road is about 240m - so how low can we go?
The answer is 0m, thanks to loops. The K5, heading towards Morden, stops at Kingston's Cromwell Road bus station, before taking a massive detour to take in the rest of the Kingston shopping district, before returning to the exact same bus stop. During the rush hour, this loop can take up to ten minutes to complete, so I'd argue this could also be the longest time between the bus departs and arrives, and you catch up.
There are other instances where we can get close - cross the street just as the 224 begins its loop around Brent Park, and you'll probably make it:
And the 359 has to get around the entirety of the Monks Hill estate before returning to the main road:
These are also more likely to prove beneficial for this question as the smaller the loop, the more likely the route sails through.
The record smallest distance, however, is 0m...and because negative distances don't exist, we'll never beat it.
Inspired by this thread on the Bus Forum.
Images courtesy of Google Maps and TfL.
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