A plane flies due north (90° from east) with a velocity of 100 km/h for 3 hours. During this time, a steady wind blows southeast at 30 km/h at an angle of 315° from due east. After 3 hours, where will the plane’s position be relative to its starting point? Show your work.

Respuesta :

consider east-west direction along x-axis with east pointing towards positive x-axis.

consider north-south direction along y-axis with north pointing towards positive y-axis

\underset{v_{p}}{\rightarrow} = velocity of the plane = 0 i + 100 j

\underset{v_{w}}{\rightarrow} = velocity of the wind = (30 Cos315) i + (30 Sin315) j = 21.2 i - 21.2 j

net velocity of the plane is given as

\underset{v_{net}}{\rightarrow} = \underset{v_{p}}{\rightarrow} + underset{v_{w}}{\rightarrow}

\underset{v_{net}}{\rightarrow} = (0 i + 100 j) + (21.2 i - 21.2 j ) = 21.2 i + 78.8 j

t = time of travel = 3 hours

position of the plane is given as

\underset{X}}{\rightarrow}  = \underset{v_{net}}{\rightarrow} t

\underset{X}}{\rightarrow}  = (21.2 i + 78.8 j ) (3)

\underset{X}}{\rightarrow}  = 63.6 i + 236.4 j

magnitude of distance from the initial starting point is given as

d = sqrt((63.6)² + (236.4)²) = 244.81 m

direction is given as

θ = tan⁻¹(236.4/63.6) = 75 deg north of east.

Answer:

244.78 km, 74.9⁰ North East

Explanation:

Step 1: identify the given parameters

Velocity of plane = 100 km/h

Velocity of wind = 30 km/h

Time of flight = 3 hours

Distance traveled by plane = (100 km/h)*(3 hours) = 300 km

Distance blew by wind = (30 km/h)*(3 hours) = 90 km

Step 2: construct a triangle with the distance of plane and wind as show in the imaged uploaded.

Step 3: calculate the final position (R) of the plane

Using cosine rule, calculate R

R² = 300² + 90² -2(300X90)cos315

R² = 98,100 - 54,000cos315

R² = 98,100 - 38,183.766

R² = 59,916.234

R = √59,916.234

R = 244.78 km

Step 4: calculate the direction of the plane(Ф)

Using sine rule, calculate the direction of the plane

[tex]\frac{90}{sine \theta} =\frac{244.78}{sine 315}[/tex]

[tex]sine\theta =\frac{90*sine315}{244.78}[/tex]

[tex]sine\theta = -0.25998[/tex]

[tex]\theta = -15.07[/tex]

[tex]\theta = 90-15.07[/tex] = 74.9⁰ North East

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