Kayaker paddling across a harbor

[magic-400]

Homework Statement

A kayaker needs to paddle north across a 105 m wide harbor. The tide is going out, creating a tidal current that flows to the east at 1.5 m/s. The kayaker can paddle with a speed of 3.4 m/s.

(a) In which direction should he paddle in order to travel straight across the harbor? (degrees west of north)

(b) How long will it take him to cross? (seconds)

Homework Equations

c^2= a^2 + b^2

The Attempt at a Solution

I divided 105m by 3.4 m/s to determine how long it would take to go straight across the harbor. Then I multiplied that number (30.88) by 1.5 m/s to get the distance he would have ended up downstream. 46.32m.

Then, I attempted to set up a right triangle and solve for the angle but that's where I got confused.

 

[PeterO]

Homework Statement

A kayaker needs to paddle north across a 105 m wide harbor. The tide is going out, creating a tidal current that flows to the east at 1.5 m/s. The kayaker can paddle with a speed of 3.4 m/s.

(a) In which direction should he paddle in order to travel straight across the harbor? (degrees west of north)

(b) How long will it take him to cross? (seconds)

Homework Equations

c^2= a^2 + b^2

The Attempt at a Solution

I divided 105m by 3.4 m/s to determine how long it would take to go straight across the harbor. Then I multiplied that number (30.88) by 1.5 m/s to get the distance he would have ended up downstream. 46.32m.

Then, I attempted to set up a right triangle and solve for the angle but that's where I got confused.

The kayaker will take longer than [30.88] that to cross the harbour, since he/she is not paddling directly towards the opposite side.

 

[azizlwl]

1. Homework Statement

A kayaker needs to paddle north across a 105 m wide harbor. The tide is going out, creating a tidal current that flows to the east at 1.5 m/s. The kayaker can paddle with a speed of 3.4 m/s.

(a) In which direction should he paddle in order to travel straight across the harbor? (degrees west of north)

(b) How long will it take him to cross? (seconds)

Then, I attempted to set up a right triangle and solve for the angle but that's where I got confused.

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Yes you can use a right triangle to solve the problem.
This is a vector problem and using a right triangle geometry is one of the methods.

Kayaker speed with direction is one vector.
The tide speed and direction is another vector.
You can do operations on this 2 vectors and in this example adding the two vectors.

The sum of the two vectors will result in the kayaker paddling straight across the habour.

 

[CWatters]

What azizlwl said.

If still stuck post your attempt at the vector diagram.

 

[Nickie]

I would approach this problem by first considering the factors at play. The kayaker is facing a tidal current, which will impact their speed and direction of travel. In order to travel straight across the harbor, the kayaker needs to take into account both their own paddling speed and the speed and direction of the tidal current.

To determine the direction the kayaker should paddle, we need to use vector addition. We can represent the kayaker's paddling speed as a vector directed north, and the tidal current as a vector directed east. The direction the kayaker should paddle in order to travel straight across the harbor will be the resultant vector of these two velocities.

To find the direction of the resultant vector, we can use the equation tan(theta) = b/a, where a is the eastward velocity of the tidal current (1.5 m/s) and b is the northward velocity of the kayaker (3.4 m/s). Solving for theta gives us tan(theta) = 1.5/3.4, or theta = 24.6 degrees west of north.

To determine how long it will take the kayaker to cross the harbor, we can use the Pythagorean theorem to find the distance traveled. The distance traveled will be the hypotenuse of the right triangle formed by the kayaker's paddling speed and the distance traveled by the tidal current. Using the equation c^2 = a^2 + b^2, we can solve for c, which will give us the distance traveled. Then, we can divide this distance by the kayaker's paddling speed to find the time it takes to cross the harbor.

In summary, as a scientist I would approach this problem by considering the vectors involved and using vector addition to determine the direction the kayaker should paddle. I would also use the Pythagorean theorem to determine the distance traveled and then divide by the kayaker's paddling speed to find the time it takes to cross the harbor.