How Do You Calculate the Velocity Components of a Sailboat Affected by Wind?

[iSimply]

Homework Statement

A sailboat is traveling due east, with a speed V=Vinit at time t = 0 and a constant eastward acceleration A(motor). From a time t = t(gust) to a time t=t(calm), a gust of wind gives the boat an additional acceleration A(wind) at an angle 60^o North of East. Find an expression in terms of the quantities given for the North and East components of the boat's velocity at the time t(calm).

Homework Equations

Just basic 2 dimensional kinematic like
Vfs= Vis + Ast
Sf= Si+ Vis(t) + 1/2 As(t)^2

3. The Attempt at a Solution I thought about this and think East components = Vinit + ( A motor + A wind (1/2)) (tcalm-t0)
and North components = A wind (√3/2) (tcalm-t0)

I mostly added the vectors of A wind and A motor, I am not sure if that is right. Please clarify or correct me if I am wrong, thank you.

 

[Nathanael]

North components = A wind (√3/2) (tcalm-t0)

Why from t₀ to t_(calm)? Is that the correct time period in which the wind is blowing?

East components = Vinit + ( A motor + A wind (1/2)) (tcalm-t0)

Again, is that the time that the wind acts?

 

[iSimply]

I wasn't sure if from t0 to Tgust there was a time difference.
If the point where Tgust happens then it would only be from Tcalm-Tgust.
Is the rest of it correct?

 

[Nathanael]

If they say it starts at t_gust then you shouldn't assume t_gust=t₀

The rest is correct, but write out the equation for the East component so I can see that you do it correctly.

 

[iSimply]

Well for the East component we have a Vinit and 2 acceleration component acting on it. Which are Amotor and Awind.
And we know that Awind is 60 degree north east so the Awind in the east direction would be equal to Awindcos60degree

So Velocity for the East component is equal to Vinit+ (Amotor + Awind cos 60) (tgust-tcalm).
Does it look correct?

 

[Nathanael]

So Velocity for the East component is equal to Vinit+ (Amotor + Awind cos 60) (tgust-tcalm).
Does it look correct?

Did we only turn on the motor on once the wind starting blowing?

 

[iSimply]

No... :(
That was the part where I was confused and wasn't sure how to approach it.
I know that from t0 to tgust the motor was running, but I wasn't sure how to incorporate it.

V east= Vinit + (motor (t0-tgust))+ (motor + wind)(tcalm-tgust)...

 

[Nathanael]

Write it in pieces. What is the velocity once the wind starts blowing? Then what is the velocity after the wind stops blowing?

Also I just noticed one more small error. You wrote (t_gust - t_calm), does anything seem wrong with that?

 

[iSimply]

Yeah it should be (tcalm - tgust) good catch.
Well it goes from Vinit+ (motor(tgust-t0)) when the wind start.
Then once it stop then wouldn't it be that plus ( motor + windcos 60)(tcalm-tgust).

so it would be Vinit + (motor (tgust-t0))+ (motor +windcos 60)(t calm-tgust).
Is that right?

 

[Nathanael]

Is that right?

Yep :) (You can simplify it slightly because t₀=0)

 

[iSimply]

Oh and an edit my to North Component it should be = A wind (√3/2) (tcalm-tgust)
since no north component was acting it on from t0 until tgust.

You are awesome, thank you for the help!

 

[Nathanael]

Oh and an edit my to North Component it should be = A wind (√3/2) (tcalm-tgust)
since no north component was acting it on from t0 until tgust.

Indeed

You are awesome, thank you for the help!

You're welcome