Respuesta :
The resultant velocity is about 12.3 feet per second at a heading of 8.4°east of north, <1.8, 12.14>
In this question,
Amy is paddling her canoe across a lake at a speed of 10 feet per second headed due north.
The wind is blowing 40°C east of north with a velocity of 2.8 feet, per second.
We need to find Amy's resultant velocity.
We write the direction vector for Amy in component form.
Due north is equivalent to 90° from the horizontal.
x-component:
x = 10 cos(90°)
x = 0
And y-component:
y = 10 sin(90°)
y = 10
The component form for Amy’s vector is <0, 10>.
Now we write the direction vector for wind in component form.
40 east of north is equivalent to 50° from the horizontal.
x-component:
x = 2.8 cos(50°)
x = 1.8
And y-component:
y = 2.8 sin(50°)
y = 2.14
The component form for the wind’s vector is <1.8, 2.14>
Add the vectors for Amy and the wind.
<0, 10> + <1.8, 2.14> = <1.8, 12.14>
Now, we determine the vector's magnitude by using Pythagoras theorem.
Let v be the velocity of Amy.
Let θ be the angle made by Amy with the horizontal
⇒ tan(θ) = |y/x|
⇒ tan(θ) = |12.14 / 1.8|
⇒ θ = 81.6°
81.6° with the horizontal is equivalent to 8.4° east of north.
Therefore, the resultant velocity is about 12.3 feet per second at a heading of 8.4°east of north, <1.8, 12.14>
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