2.Find the magnitude and direction of the resultant velocity vector for the following perpendicular velocities: a. a fish swimming at 3.0 m/s relative to the water across a river that moves at 5.0 m/s b. a surfer traveling at 1.0 m/s relative to the water across a wave that is traveling at 6.0 m/s

a. The magnitude of the resultant velocity vector of the fish is approximately 5.83 m/s

The direction of the resultant velocity vector of the fish is approximately 30.93° relative to the direction of the river

b. The magnitude of the resultant velocity vector of the surfer is approximately 6.083 m/s

The direction of the resultant velocity vector of the surfer is approximately 9.46° relative to the direction of the wave

Explanation:

a. The given parameters of the fish and the river are;

The speed with which the fish is swimming across the river, v₁ = 3.0 m/s

The velocity  with which the river is flowing, v₂ = 5.0 m/s

Therefore, in vector form, we have;

v₁ = 3.0·j

v₂ = 5.0·i

The resultant velocity, R = v₁ + v₂ = 5.0·i + 3.0·j

∴ R = 5.0·i + 3.0·j

The magnitude of the resultant velocity vector, = √(5.0² + 3.0²) =√34 ≈ 5.83

The magnitude of the resultant velocity vector,  of the fish,    ≈ 5.83 m/s

The direction of the resultant velocity of the fish, θ = arctan(3.0/5.0) ≈ 30.93°

Therefore, The direction of the resultant velocity vector of the fish, θ ≈ 30.93° relative to the direction of the river

b. The given parameters of the surfer and the wave are;

The speed with which the surfer is travelling, v₁ = 1.0 m/s

The speed of the wave travelling across, v₂ = 6.0 m/s

Therefore, in vector form, we have;

v₁ = 1.0·j

v₂ = 6.0·i

The resultant velocity, R = v₁ + v₂ = 1.0·i + 6.0·j

∴ R = 1.0·i + 6.0·j

The magnitude of the resultant velocity of the surfer, = √(1.0² + 6.0²) =√37 ≈ 6.083

The magnitude of the resultant velocity vector of the surfer,  ≈ 6.083 m/s

The direction of the resultant velocity of the surfer, θ = arctan(1.0/6.0) ≈ 9.46°

Therefore, The direction of the resultant velocity vector of the surfer, θ ≈ 9.46° relative to the direction of the wave

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