in this video we're studying relative velocities and we start with a river that has a three and a
half meter per second current to the east and then we have a boat that can move at seven and a half
meters per second with respect to the water and it's pointed straight to the north pushing the
boat seven and a half meters per second with respect to that water the river is 70 meters
across and in part a of the problem we're going to find the velocity of the boat with respect to
the shore and in part b of the problem we're going to find this eastward displacement delta x as the
boat crosses the river aiming exactly at that opposite shore but getting dragged to the east
by the current and so what we've learned about relative velocities is they simply add if you do
it in the right order and i can say the velocity of the boat with respect to the shore this is kind
of an informal notation for this i write v with a subscript of boat slash shore the velocity of the
boat with respect to the shore is equal to the velocity of the boat with respect to the water
plus the velocity of the water with respect to the shore and the velocity of the boat with respect
to the water that's seven and a half meters per second pointed to the north and the velocity of
the water with respect to the shore that's three and a half meters per second pointed to the east
so to vector add these we're going to need a triangle and i've got my seven and a half
meters per second to the north my three and a half meters per second to the east
and then what i'm interested in here is the total velocity of the boat with respect to the shore
in polar form and we can just measure that angle with respect to the northward direction
so the magnitude of that total velocity is given by the pythagorean theorem and when we run the
numbers on this we get 8.28 meters per second with respect to the shore the angle is given by an
inverse tangent so this is the angle whose tangent is three and a half that's the opposite side
over seven and a half the adjacent side and when we run the numbers on this we get 25.0 degrees
again sticking with the convention of just keeping three sig figs on everything
now if you prefer you could label that angle with respect to the eastward
direction instead i'll call that phi and of course theta and phi add to 90 degrees
so it's also fine to express it as an angle phi with respect to the east and that would be 65
degrees in part b we want to find the eastward displacement of the boat when it arrives at the
opposite shore and there's two different ways to do this i think the quickest way to do it is
to say well if the boat is always moving at 25 degrees with respect to the northward direction
and we know that it moves 70 meters in the northward direction
we can just sketch a triangle for the displacement that's 70 meters to the north some unknown delta x
for the eastward displacement and i know that it happens at an angle of 25 degrees with respect to
the northward direction this means i can just use a tangent function to get access to that delta x
so i write down the tangent of 25 degrees is delta x over 70 that's the opposite over adjacent
so delta x is 70 tangent 25 and when i run the numbers on this i get 32.6 meters
now there's another approach to part b and i think it's instructive to show this if i really want to
get into the nitty-gritty of the kinematics here i can find the time for the boat to cross the river
and then i can use its eastward velocity that three and a half meter per second eastward current
to figure out what the eastward displacement is so first to find the time to cross the river
i just have a simple relationship of distance equals rate times time so i'm going to write y
equals v y times t that means the time to cross the river is y over v y and it covered 70 meters
in that y direction at a speed of 7.5 meters per second in that y direction
and this gives me a crossing time of 9.33 seconds then we look at the horizontal
direction and to describe the kinematics in the horizontal direction again it's moving at
a constant velocity in that direction so it's really just distance equals rate times time
x equals the x velocity times the time and that current was moving to the east at three and a
half meters per second and it did that for 9.33 seconds as the boat crosses the river
and when i run the numbers on this i get 32.7 meters so there's a slight discrepancy between
the two answers one was 32.6 and one was 32.7 that can be attributed to rounding error and we
finish with a high degree of confidence that we've got the right answer if you enjoyed this video or
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