Calculating Speed and Direction of Sailboat After Gust of Wind

[Kster]

A sailboat is traveling east at 5m/s . A sudden gust of wind gives the boat an acceleration=.80 m/s^2, (40 degrees north of east).

A. What is the boat's speed 6 seconds later when the gust subsides?
________m/s

B. What is the boat's direction 6 seconds later when the gust subsides?
________degrees north of east.


My attempts:
v = v0 + at
v = (0) + (0.80m/s^2)* (6sec)
v = 4.8m/s

A^2+ B^2 = C^2
(4.8)^2 + (5.0)^2 = 48.04
C= 6.9 m/s
_____________________

tan-1 (4.8/5.0) = 44°


Please tell me what I did wrong? I am so stuck :(

 

[Doc Al]

A^2+ B^2 = C^2
(4.8)^2 + (5.0)^2 = 48.04
C= 6.9 m/s

This assumes that the acceleration is perpendicular to the original velocity (east). But it's not: the acceleration is 40 degrees north of east.

Hint: Break the acceleration (and the velocity) into components (east and north). Find the final velocity component east and the final velocity component north, then you can combine them to find the final speed.

Another approach is to find the change in velocity along the direction of the acceleration, which will be some vector 40 degrees north of east. Then just add that vector to the initial velocity vector and find the new magnitude.

 

[Moetasim]

Firstly, for part A, you have correctly calculated the boat's speed after 6 seconds. However, you have not answered the question, which asks for the boat's speed when the gust subsides. This means that you need to add the initial speed of 5m/s to the final speed of 4.8m/s to get the total speed when the gust subsides.

Therefore, the boat's speed 6 seconds later when the gust subsides is 9.8m/s.

For part B, you have used the Pythagorean theorem to calculate the hypotenuse (C) of the triangle formed by the initial velocity and the acceleration due to the gust. However, this does not give you the direction of the boat. To find the direction, you can use trigonometric functions.

Firstly, we need to find the angle (θ) between the initial velocity and the final velocity. To do this, we can use the dot product formula:

v1 · v2 = |v1| * |v2| * cosθ

where v1 and v2 are the initial and final velocities, respectively.

We know that the initial velocity is 5m/s and the final velocity is 4.8m/s. So, the dot product becomes:

(5)(4.8) = (5)(4.8) * cosθ

cosθ = (5)(4.8) / (5)(4.8)

cosθ = 1

θ = cos^-1(1)

θ = 0°

This means that the angle between the initial and final velocity is 0°, which also happens to be the angle between the initial velocity and the direction of the gust (40° north of east).

To find the boat's direction, we need to add 40° to the initial direction of east. Therefore, the boat's direction 6 seconds later when the gust subsides is 40° north of east.

I hope this helps! Let me know if you have any further questions.