Solving Relative Position Problem: Velocity of Boat w.r.t Water

[brad sue]

Hi this problem confuses me :

A sailor wants to travels due east at a velocity of 15km/h with respect to a coordinate system fixed to the ground.
The sailor must contend with the gulf stream, which moves north at 5 km/h. With which veolcity with respect to the water should the sailboat proceed under sail?

to try solve it I set :

V_bg= velocity of boat relative to ground.

V_wg= velocity of water relative to ground.

V_bw= velocity of boat relative to water.

now I wrote: V_bg= V_bw+V_wg but For me it does not fit.
What would fit is :
V_bw= V_bg+V_gw

here V_gw is not defined in the problem!

please can someone can help me?

Also, to find the angle θ of the boat, I am confused about what ( and where)put in the equation θ =tan^-1(y/x)

Thank you

brad

 

[Andrew Mason]

Hi this problem confuses me :

A sailor wants to travels due east at a velocity of 15km/h with respect to a coordinate system fixed to the ground.
The sailor must contend with the gulf stream, which moves north at 5 km/h. With which veolcity with respect to the water should the sailboat proceed under sail?

to try solve it I set :

V_bg= velocity of boat relative to ground.

V_wg= velocity of water relative to ground.

V_bw= velocity of boat relative to water.

now I wrote: V_bg= V_bw+V_wg but For me it does not fit.
What would fit is :
V_bw= V_bg+V_gw

here V_gw is not defined in the problem!

please can someone can help me?

Also, to find the angle θ of the boat, I am confused about what ( and where)put in the equation θ =tan^-1(y/x)

Thank you

brad

The vector sum of the boat's velocity relative to the water and the water's velocity relative to the land gives you the boat's velocity relative to the land.

You know that this vector sum = 15 km/hr east. You also know that the water's velocity wrt land is 5 km/hr north. Work out the velocity of the boat relative to the water from that. $\theta$ is the angle of the boat's heading. The tangent of this angle is the north component/east component.

AM

 

[quicknote]

Hello Brad,

I can understand why this problem may be confusing for you. Let me try to break it down and provide some guidance.

First, let's define the variables:

Vbg = velocity of boat relative to ground
Vwg = velocity of water relative to ground
Vbw = velocity of boat relative to water

The key to solving this problem is understanding that the velocity of the boat relative to the water is the combination of the velocity of the boat relative to the ground and the velocity of the water relative to the ground. This can be represented as:

Vbw = Vbg + Vwg

Now, we know that the boat is traveling due east at a velocity of 15 km/h with respect to the ground. This means that Vbg = 15 km/h. We also know that the gulf stream is moving north at 5 km/h, so Vwg = 5 km/h.

Substituting these values into the equation, we get:

Vbw = 15 km/h + 5 km/h = 20 km/h

This means that the boat must travel at a velocity of 20 km/h relative to the water in order to maintain a velocity of 15 km/h relative to the ground.

As for finding the angle θ of the boat, we can use the formula θ = tan-1(y/x), where y is the vertical component of the velocity and x is the horizontal component of the velocity. In this case, y = 5 km/h (since the boat is traveling north) and x = 15 km/h (since the boat is traveling east). This gives us:

θ = tan-1(5/15) ≈ 18.4°

So, the boat must travel at a velocity of 20 km/h at an angle of 18.4° in order to maintain a velocity of 15 km/h with respect to the ground.

I hope this helps clarify the problem for you. If you have any further questions, feel free to ask. Good luck!