Question

1)По озеру движется катер со скоростью, модуль которой равен 10 м/c. К катеру прикреплен трос, за другой конец которого держится спортсмен на водных лыжах. Скорость катера направлена под углом 30 градусов к тросу, а скорость спортсмена - под углом 45 градусов к тросу. Определите модуль скорости спортсмена. 

Answer 1

Problem Analysis

We are given that a boat is moving across a lake with a speed of 10 m/s. A rope is attached to the boat, and a water skier is holding onto the other end of the rope. The boat's velocity is at an angle of 30 degrees with respect to the rope, and the skier's velocity is at an angle of 45 degrees with respect to the rope. We need to determine the magnitude of the skier's velocity.

Solution

To solve this problem, we can break down the velocities into their x and y components and then add them up to find the resultant velocity.

Let's assume the x-axis is parallel to the direction of the boat's velocity and the y-axis is perpendicular to it.

The boat's velocity can be broken down as follows: - The x-component of the boat's velocity is given by: Vbx = Vb * cos(30°). - The y-component of the boat's velocity is given by: Vby = Vb * sin(30°).

Similarly, the skier's velocity can be broken down as follows: - The x-component of the skier's velocity is given by: Vsx = Vs * cos(45°). - The y-component of the skier's velocity is given by: Vsy = Vs * sin(45°).

Since the skier is holding onto the rope, the x-component of the skier's velocity must be equal to the x-component of the boat's velocity. Therefore, we have:

Vbx = Vsx

Substituting the values, we get:

Vb * cos(30°) = Vs * cos(45°)

Simplifying the equation, we find:

Vs = (Vb * cos(30°)) / cos(45°)

Now, we can substitute the given values to find the magnitude of the skier's velocity.

Calculation

Let's calculate the magnitude of the skier's velocity using the given values: - Boat's velocity (Vb) = 10 m/s - Angle between the boat's velocity and the rope (θb) = 30° - Angle between the skier's velocity and the rope (θs) = 45°

Substituting these values into the equation, we get:

Vs = (10 * cos(30°)) / cos(45°)

Using a calculator, we can evaluate this expression to find the magnitude of the skier's velocity.

Answer

The magnitude of the skier's velocity is approximately 7.07 m/s.

Conclusion

In this problem, we determined the magnitude of the skier's velocity by breaking down the boat's and skier's velocities into their x and y components and then equating the x-components. By substituting the given values into the equation, we found that the magnitude of the skier's velocity is approximately 7.07 m/s.