1. Question
Does the current affect the time it takes a boat to cross a river if the boat is pointed directly across the river?

2. Independence of Perpendicular Vectors
Fact: Vector quantities acting on the same object, but in perpendicular directions, do not affect each other.
Think back to the Paper River Lab…
Did the river’s current make the boat’s motor go at a different speed?
Did the boat’s motor make the river’s current go at a different speed?
Note, these are not the same questions as:
Did the river’s current make the boat follow a different path than the motor would have alone?

3. Some think the current will add time to crossing the river, since the boat obviously travels a longer distance…
Others think the current will lessen the time to cross the river, since the current makes the boat move faster than with the motor alone…
In a sense, both of these ideas have merit,
but acting together,
basically cancel out each other’s effects… watch...

4. Having the current there does affect the boat’s path…
The current clearly acts on the boat in the downstream direction, forcing the boat to end up DS from where it started.
And the current does affect the boat’s resultant speed…
Not the speed the motor moves it, but the resultant speed from combining motor & current…
So the extra distance the boat is moved because there is a current, is made up for by the current also adding proportionally extra speed to the boat to carry it that extra distance. (This is an OK explanation.)

5. If the width of the river doesn’t change,
and the boat’s motor speed doesn’t change,
then the time it takes the boat to cross the river
shouldn’t change,
as long as you point your boat directly across the river!
Because by pointing your boat directly across the river, all of the boat’s velocity – the entire vector of it – is being used to get the boat across the river. (Better explanation)
If the current is acting perpendicular to the boat’s motor, it won’t affect what the motor is doing for the boat at all – no help, no hurt. So the boat will take the same amount of time to cross still water, as a river of equal distance! (Best explanation)
This is the idea of Independence of Perpendicular Vectors:
Vectors that act perpendicular to each other, do not affect each other – no help, no hurt – it’s as if the one vector isn’t even there, when figuring out what’s going on in the perpendicular direction.

6. How Could You Have Seen This in the Paper River Lab?
Consider Proc 2b, where you determined the % difference between the longest and shortest time to cross the pond.
Consider Proc 5b, where you determined the % difference between the longest and shortest time to cross the river.
What were your % differences for these 2 calculations?
For the vast majority of lab groups, they turn out to be quite large… sometimes over 20%...
Recall your answer to Q3 – in a perfect world, what should these % differences have been?
Zero, right? Because all the times for Proc 1 were really of the exact same event, so they should have been exactly the same.
And same with Proc 4. But then why were they so different? They were different due to common human error
inconsistences with the stopwatch mostly…

7. Now consider Proc 6, where you determined the % difference between the average times across the pond vs the average times across the river.
How did that work out???
For the vast majority of lab groups, they turn out to be quite small… always under 10%... (unless there was a problem)
So by science standards, these values are considered to be within an acceptable margin of error, …
and therefore, within reason, the same!!
So you should have concluded that …
the current of the river did not affect the time your boat took to cross it, as long as the boat was perpendicular to the flow of the current.

8. But what if you don’t point your boat directly across the river to get to the other side?
If you point it angled toward up or downstream,
Then the 2 velocities are no longer perpendicular to each other
Which means:
no longer is all the boat motor’s velocity working to get it directly across the river.

9. When you point the boat at an angle from directly across,
now only a COMPONENT of the boat motor’s velocity is acting ACROSS the river, while the other component is acting either directly up or down stream.
you have less velocity acting in the across direction
but still the same distance (width of the river) to cross,
so, it will take longer to make it to the other shore
pointed either angled up or angled downstream!

10. Notice this explanation has nothing to
do with any extra distance traveled.
In fact, for a boat pointed angled upstream,
it will likely travel a smaller distance to
reach the opposite side of the river!
But still, traveling this shorter distance, and landing directly across a river, will take longer , because you’re moving in the across
direction with only a component of the original speed (so less) than when you point the boat directly across the river.
Note: if you want to land directly across the river from where you started, you need to angle your boat upstream so that
its upstream component equals the speed of the current.
Then they will cancel, leaving only the across component to get you to the other shore. Odd result: the boat will be angled upstream, but actually be traveling straight across – weird!