a bird can fly in still air at 20 miles per hour. when the wind is blowing straight from the north at 10 miles/hr. in what direction must the bird aim in order that it may fly from east to west and at what speed will it have?

20^2 = 10^2 + v^2

sin(Θ) = 10/20 ... north of west

To determine the direction and speed at which the bird must aim in order to fly from east to west, we need to consider the relative motion of the bird and the wind.

First, let's break down the motion:

1. Bird's airspeed: The bird can fly at 20 miles per hour in still air. This is its airspeed, unaffected by the wind.

2. Wind speed: The wind is blowing straight from the north at 10 miles per hour.

To fly from east to west, the bird needs to counteract the effect of the wind blowing from the north.

Now, let's calculate the bird's required groundspeed and direction:

1. Groundspeed: The bird's groundspeed is the combination of its airspeed and the wind speed. Since the wind is coming from the north (opposite to the desired east-west direction), the bird needs to compensate for the wind's effect.

The bird's groundspeed should be the vector sum of its airspeed and the opposite vector of the wind speed. So, we subtract the wind speed from the bird's airspeed:

Groundspeed = Bird's airspeed - Wind speed
Groundspeed = 20 miles/hour - 10 miles/hour
Groundspeed = 10 miles/hour

2. Direction: The bird needs to aim in a direction that compensates for the effect of the wind. Since the wind is coming from the north, the bird needs to fly slightly south of west to counteract the wind's northward push.

To determine the angle at which the bird should aim, we can use the tangent of the angle (opposite/adjacent) in a right-angled triangle. In this case, the opposite side is the wind speed (10 mph), and the adjacent side is the bird's airspeed (20 mph). So:

Tangent of the angle = opposite/adjacent
Tangent of the angle = 10/20
Tangent of the angle = 0.5

Taking the arctangent (inverse tangent) of 0.5, we get the angle:

Angle = arctan(0.5)
Angle ≈ 26.57 degrees

Therefore, the bird must aim approximately 26.57 degrees south of west and maintain a groundspeed of 10 miles per hour in order to fly from east to west.