Relative Velocity: Al & Bob's Trip to Ted's Place

AI Thread Summary
Al and Bob, two swimmers, need to coordinate their trips across a 200 m wide river to reach their friend Ted simultaneously. Al's travel time is calculated at approximately 142.85 seconds, while Bob's direct route, factoring in the river's current, takes about 128.20 seconds. To synchronize their arrivals, Al must wait 14.62 seconds after Bob starts swimming. The discussion highlights the importance of understanding vector addition in determining Bob's swimming angle against the current. Correctly calculating their velocities and angles is crucial for both swimmers to arrive at Ted's place together.
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Two swimmers, Al and Bob, live on opposite shores of a 200.0 m wide river that flows east at 0.70 m/s. Al lives on the north shore and Bob lives on the south shore. They both set out to visit a mutual friend, Ted, who lives on the north shore at a point 100.0 m upstream from Al and 100.0 m downstream from Bob. Both swimmers can swim at 1.4 m/s through the water. How much time must Al wait after Bob sets out so that they both arrive at Ted's place at the same time? Both swimmers make their trips by the most direct routes.

So, first I can find the time taken by Al.
t=D/V_ag
= 100/V_aw-V_wg
=100/0.7= 142.85 Seconds

Now, for Bob

we could use trig to find the angle (beta) of his most direct route.
but 1st we need to find theta.

So using tri==== Tan(theta)= 200/100
And we get theta as 63.4 degrees.
Then we can find beta, since one angle is 90 and the other is 63.4.
so beta will be= 26.5 degrees
Then, Sin(26.5)= 0.7/hypotunese
hence hypotunese= 1.56m/s
This is bob's speed (most direct route)
Now for his time, t= 200/1.56= 128.20 seconds.

T1-T2= 142.82-128.20= 14.62 seconds.
hence Al must wait for 14.62 seconds after Bob has left inorder for both of them to reach Ted's place at the same time.

Does this look good?
 
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Your answer for Al is correct but you need to show that he can reach the other bank in that time, too.

Regards Bob, the angle his most direct route makes with the banks is your theta and the angle it makes perpendicular to the banks is your beta (you miscalculate beta by 0.01 degrees).

Bob's "most direct route" diagram can't be used as the vector addition triangle to find the velocity that he travels on that route. Imagine yourself as Bob, swimming. What direction would you have point your body so that addding your swimming velocity to the river's flow velocity, you travel on the most direct route?
 
I guess, Bob will have to travel towards the north direction, and then let the eastwards water current carry him.
 

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