Final answer:
The velocity of the wind relative to the cyclist, who is traveling due east at 12 km/h, with the wind blowing from the south-west at 5 km/h, can be calculated using vector addition. The wind's component that is relevant to the cyclist is the one perpendicular to their direction of movement, yielding an effective north-east velocity of approximately 3.54 km/h.
Step-by-step explanation:
To find the velocity of the wind relative to the cyclist who is traveling due east at 12 km/h, with the wind blowing from the south-west at 5 km/h, we need to use vector addition. The wind has a component blowing towards the north-east (opposite to south-west direction) and another component blowing towards the south-east. However, since the cyclist is moving east, we only need to consider the component of the wind's velocity that is perpendicular to the cyclist's velocity, which is blowing from the south-west towards the north-east.
Using vector addition:
- Break down the wind velocity into two perpendicular components: one in the direction of the cyclist's travel (eastward) and the other perpendicular to it (northward).
- Because the wind direction is from the south-west, these components have equal magnitude since south-west is 45 degrees off of both the south and west axes.
- The eastward wind component is canceled out by the cyclist's travel, so we focus on the northward component.
- Since the angle is 45 degrees, and using basic trigonometry, the northward (and eastward) component of the wind's velocity is 5 km/h * cos(45 degrees) = 5 km/h * 0.7071 ≈ 3.54 km/h.
Therefore, the cyclist feels the wind coming from the side, from the south-west, with an effective velocity of 3.54 km/h to the north-east relative to their movement.