To determine the angle at which the raindrops appear to fall from the perspective of the motorcyclist, we need to analyze the velocities involved. The motorcyclist is moving towards the east at a constant speed of 15 m/s, while the raindrops are falling vertically with a velocity of 20 m/s at a 10-degree angle from the vertical. We can break this problem down into components and use some trigonometry to find the angle of the raindrops as perceived by the motorcyclist.
Understanding the Components of Motion
First, let's resolve the velocity of the raindrops into its horizontal and vertical components. The vertical component (Vry) can be calculated using the cosine of the angle, and the horizontal component (Vrx) can be calculated using the sine of the angle:
- Vertical Component (Vry): Vry = Vr * cos(θ) = 20 m/s * cos(10°)
- Horizontal Component (Vrx): Vrx = Vr * sin(θ) = 20 m/s * sin(10°)
Calculating these components:
- Vry = 20 m/s * cos(10°) ≈ 19.7 m/s
- Vrx = 20 m/s * sin(10°) ≈ 3.5 m/s
Relative Velocity of Raindrops
Next, we need to find the relative velocity of the raindrops as seen by the motorcyclist. Since the motorcyclist is moving east at 15 m/s, we can subtract the horizontal component of the raindrop's velocity from the motorcyclist's velocity:
- Relative Horizontal Velocity (Vr_x): Vr_x = Vrx - Vmotorcyclist = 3.5 m/s - 15 m/s = -11.5 m/s
- Relative Vertical Velocity (Vr_y): Vr_y = Vry = 19.7 m/s
The negative sign indicates that the relative horizontal velocity is directed opposite to the motorcyclist's motion, which is expected since the raindrops are falling towards the ground while the motorcyclist is moving east.
Calculating the Apparent Angle
Now, we can find the angle at which the raindrops appear to fall from the motorcyclist's perspective using the arctangent function:
- Apparent Angle (φ): φ = arctan(Vr_y / |Vr_x|) = arctan(19.7 m/s / 11.5 m/s)
Calculating this gives:
- φ ≈ arctan(1.713) ≈ 59.7 degrees
Final Thoughts
Thus, the angle at which the raindrops would appear to fall as seen by the motorcyclist is approximately 59.7 degrees from the vertical. This means that while the raindrops are actually falling at a 10-degree angle from the vertical, due to the motion of the motorcyclist, they appear to be coming from a much steeper angle. This example illustrates how relative motion can significantly alter our perception of direction and speed in different frames of reference.
