Where does RoS and length contraction and time dilation come into play with the little boat?
The amount is negligible. It's implied in the problem, since the speeds given are nowhere near light speed. This is what we mean by calling it a problem in Galilean relativity - it's just concerned with the change in coordinates from one reference from to another, not with the spacetime warp that accompanies all motion.
But that doesn't mean we can't arrive at an answer. It's just silly because the amounts are negligible.
To see what I mean by negligible, you need only understand that in the spacetime warp question, also known as the Lorentz transformation, there is a ratio v/c, where v is any of the speeds in the problem and c is the speed of light.
The speed of light is 300,000,000 m/s.
What's the speed of the objects in question in m/s? Here I'll refer back to the way the problem stood in the OP:
Convert km/hr to m/s
3.27 km/hr = 3,270 m / 3,600 s = 0.9083 m/s (velocity of the water relative to the shore)
8 km/hr = 8,000 m / 3,600 s = 2.222 m/s (velocity of the boat relative to the water)
11.27 km/hr = 11,270 m / 3,600 s = 3.131 m/s (velocity of the boat relative to the shore as both boat and water approach shore)
Already we have an exceedingly negligible speeds but let's go further to calculate v/c:
0.9083 / 300,000,000 = 0.000000003120 m/s
2.222 / 300,000,000 = 0.000000007407 m/s
3.131 / 300,000,000 = 0.00000001044 m/s
I skipped the preferred scientific notation to emphasize scale. By the way, that last number is 1.044 nano-meters (billionths of a meter) per second. That's about 3 water molecules per second difference from saying 11.27 km/hr. And we haven't finished the calculation yet.
Here I'll go ahead and switch back to scientific notation and give you the results. You won't be able to solve this without at least 34 digits of precision - that's how negligible the difference is, and the reason we drop it in almost all cases like this. Here, I'm using a 34 digit calculator:
sqrt(1-((3270/3600)/3E8)) =0.999999998486 (1)
sqrt(1-((8000/3600)/3E8)) = 0.999999996296 (2)
sqrt(1-((11270/3600)/3E8)) = 0.999999994782 (3)
(1) gives you the length contraction for the moving water, relative to the shore. It amounts to about 5 water molecules per meter of the water, as a meter would be measured in the water's reference frame.
(2) gives you the length contraction for the boat moving at 8 km/hr, relative to the shore. It amounts to a whopping 12 molecules .
(3) gives you the length contraction for the boat approaching the shore, moving at 8 km/hr relative to the water, as the water approached the shore at 3.27 km/hr. This gives you a jumbo reduction in length of 17 molecules.
Same goes for redshift in speed. And the reciprocal is true for time dilation ( "1 divided by" the results above [1/sqrt(...etc]):
(1) yields about 1.5 ns per second of clock time, by which time dilates on the water's surface, relative to the shore, as it moves toward the shore.
(2) yields about 3.7 ns per second of clock time, by which time dilates on the boat, relative to the moving water (or to the shore in still water) as the moves at 8 km/hr relative to the water
(3) yields about 5.2 ns per second of clock time, by which time dilates on the boat, relative to the shore - as both the boat and the water move toward the shore along a straight line, with the boat moving at 8 km/hr relative to the water and the water moves at 3.27 km/hr relative to the shore.
Notice (1) + (2) = (3) in both length contraction and time dilation. All the more reason we say that the velocity of A with respect to B plus the velocity of B with respect to C equals the velocity of A with respect to C, and it applies equally to length contraction and time dilation.
(I used the numbers from the original problem, with just the first amendment - eliminating the 5 km/hr speed upstream, and imposing a 5 km distance in each leg. I didn't pay attention to the numbers used in the toy boat example, but you can work this out now if you like. It will give you exceedingly small numbers, given the fact that the toy boat can't move as fast as a motor boat.)
Take a stab at it. Make sure you use a calculator with enough digits. Go for it, man, the whole cast of readers is anxiously waiting for MotorDaddy to have a breakthrough.
