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The velocity of the current in a river has no effect on the the time it takes to paddle a canoe across the river, given that the boat is pointed perpendicular to the bank of the river, because
The velocity of the river does not change the velocity and therefore the distance traveled in the direction of the boat which is directed perpendicular to it
Reason:
Let represent the velocity of the boat across the river in the direction,
, and let,
, represent the velocity of the river, we have;
The velocity of the boat perpendicular to the direction of the river =
Therefore, the distance covered per unit time in the perpendicular direction
to the flow of the river is , such that the time it takes to cross the river in
the perpendicular direction is the same, for every value of the velocity of
the river.
This is so because the velocity in the perpendicular direction to the flow of
the river, which is the velocity of the boat is unchanged by the velocity of
the river, because there is no perpendicular component of velocity in the
velocity of the river.
= 3 m/s
= 4 m/s
The width of the river, = 6 meters, we have;
- The resultant velocity =
The direction, θ, is given as follows;
The length of the path of the boat, <em>l</em>, is given as follows;
The length of the path the boat takes = 10 m
The time it takes to cross the river, t = , therefore;
- The time it takes to cross the river, t = 2 seconds
Considering only the y-components, we have;
Therefore;
Which expresses that the time taken is the same and given that the
vectors of the velocities of the river and the boat are perpendicular, the
distance covered in the direction of the boat is unaffected by the velocity
of the river.
Learn more vectors here: