Speed And Velocity

Step 1: Understanding the formula for wave speed.
The wave speed of the ultrasound in the medium can be found using the formula: The signal travels to the depth and back, so the total distance is .

Step 2: Calculating the wave speed.
The time taken for the wave to travel this distance is seconds. Therefore, the wave speed is:

Step 3: Finding the wavelength.
The wavelength is related to the wave speed and frequency by the formula: Substitute the known values: Solving for :

Step 4: Conclusion.
Thus, the wavelength of the ultrasound signal in that medium is 2 cm.

To solve this problem, we use the equations of motion for a vertically thrown object under gravity. Initially, we apply the kinematic equation: v² = u² - 2gh, where v is the final velocity, u is the initial velocity, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the maximum height.

Given: Initial velocity v₁ makes the ball reach a height h = 12 m with final speed v = 12 m/s. Applying the equation:

12² = v₁² - 2 * 9.81 * 12

144 = v₁² - 235.44

v₁² = 379.44

v₁ = √379.44 ≈ 19.48 m/s

Next, considering the initial velocity v₂ for which the ball reaches exactly 12 m with zero final velocity, we use:

0 = v₂² - 2 * 9.81 * 12

v₂² = 235.44

v₂ = √235.44 ≈ 15.35 m/s

Finally, we calculate the ratio v₁/v₂ as:

v₁/v₂ = 19.48 / 15.35 ≈ 1.27

To determine how a person should hold their umbrella, let's analyze the situation considering both the rain and wind speeds as vector components.

1. Rain is falling vertically, so it has a vertical speed, .
2. Wind blows horizontally from west to east, hence it has a horizontal speed, .
3. The resultant velocity of the rain considering both components (vertical fall and horizontal wind) can be calculated using vector addition. This resultant will determine the direction in which the person should hold the umbrella.

To find the angle with respect to the vertical, use the tangent function:

Substitute the given values:

Now calculate :

Therefore, the person should hold the umbrella at an angle of approximately with the vertical. As the wind is blowing from west to east, the umbrella should be tilted towards the west to counteract the effect of the wind.

Hence, the correct answer is:

At an angle of about 22° with vertical towards west