a duck is swimming across a river.The duck can swim at maximum speed of 2m\s in still water. If the duck is swimming as hard as possible straight across a river that is running at 4m\s, what is the duck's overall speed, relative to the goose sitting on the river boat?

what is sqrt(4^2+2^2) ?

To determine the duck's overall speed relative to the goose, we need to consider the vector addition of the duck's swimming speed and the river's current.

Given:
- Duck's maximum swimming speed in still water = 2 m/s
- River's current speed = 4 m/s

To find the duck's overall speed, we can use the concept of vector addition. Let's break down the duck's velocity into two components:
1. The horizontal component (across the river)
2. The vertical component (upstream or downstream)

1. Horizontal Component:
Since the duck is swimming straight across the river, the horizontal component of the duck's swimming speed is equal to the maximum swimming speed in still water. Therefore, the horizontal component remains 2 m/s.

2. Vertical Component:
The vertical component of the duck's velocity depends on the river's current. If the duck is swimming towards the opposite shore, it needs to counteract the downstream current. Hence, the vertical component is the same as the river's current speed but in the opposite direction. Thus, the vertical component is -4 m/s.

Using the Pythagorean theorem, we can calculate the duck's overall speed. The Pythagorean theorem states that the square of the hypotenuse (overall speed) is equal to the sum of the squares of the other two sides (horizontal and vertical components).

Overall speed = √(horizontal component^2 + vertical component^2)
Overall speed = √(2^2 + (-4)^2)
Overall speed = √(4 + 16)
Overall speed = √20
Overall speed ≈ 4.47 m/s

Therefore, the duck's overall speed relative to the goose is approximately 4.47 m/s.

To find the duck's overall speed relative to the goose, we need to consider the velocity vectors of both the duck and the river.

The duck can swim at a maximum speed of 2 m/s in still water. This means that if there were no river current, the duck would be able to swim directly across the river at a speed of 2 m/s.

However, there is a river current flowing at 4 m/s. This means that the duck is also being pushed downstream by the current as it swims across the river.

To determine the duck's overall speed, we need to consider the vector addition of the duck's swimming speed and the river current.

Since the duck is swimming perpendicular to the direction of the current, their velocities can be added geometrically.

Using the Pythagorean theorem, we can calculate the overall speed of the duck:

Overall Speed^2 = Duck's Swimming Speed^2 + River Current Speed^2

Overall Speed^2 = (2 m/s)^2 + (4 m/s)^2

Overall Speed^2 = 4 m^2/s^2 + 16 m^2/s^2

Overall Speed^2 = 20 m^2/s^2

Taking the square root of both sides gives us:

Overall Speed = √20 m/s

Therefore, the duck's overall speed, relative to the goose, is approximately 4.47 m/s.