To find the resultant velocity of the van relative to the Earth during the gust, we can use vector addition. Let's break down the given velocities into their horizontal (x) and vertical (y) components:
Velocity of the van: 30 m/s at 20 degrees south of east
The x-component of the van's velocity can be found using cosine:
Vx_van = 30 m/s * cos(20°)
The y-component of the van's velocity can be found using sine:
Vy_van = 30 m/s * sin(20°)
Velocity of the wind: 20 m/s due east
Since the wind is blowing due east, its y-component is zero, and its x-component is 20 m/s.
To find the resultant velocity, we add the respective components of the van's velocity and the wind's velocity:
Resultant x-component: Vx_resultant = Vx_van + Vx_wind Resultant y-component: Vy_resultant = Vy_van + Vy_wind
Substituting the values:
Vx_resultant = Vx_van + Vx_wind = 30 m/s * cos(20°) + 20 m/s
Vy_resultant = Vy_van + Vy_wind = 30 m/s * sin(20°) + 0
Calculating the values:
Vx_resultant ≈ 30 m/s * 0.9397 + 20 m/s ≈ 28.19 m/s + 20 m/s ≈ 48.19 m/s
Vy_resultant ≈ 30 m/s * 0.3420 + 0 ≈ 10.26 m/s
Therefore, the resultant velocity of the van, relative to the Earth during the gust, is approximately 48.19 m/s east of the van's original direction, with a vertical component of 10.26 m/s.