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Chapter 3: Problem 97

A beautiful girl is going eastwards with a velocity of 4 \(\mathrm{km} / \mathrm{hr}\). The wind appears to blow directly from the north. She doubles her speed and the wind seems to come from north east. The actual velocity of wind is : (a) \(4 \sqrt{2} \mathrm{~km} / \mathrm{hr}\) towards south east (b) \(4 \sqrt{2} \mathrm{~km} / \mathrm{hr}\) towards north west (c) \(2 \sqrt{2} \mathrm{~km} / \mathrm{hr}\) towards south east (d) none of the above

Short Answer

Expert verified
The wind's velocity is \(4\sqrt{2}\, \mathrm{km/hr}\) towards the southeast (option a).

Step by step solution

01

Define the Variables

Let the girl's initial velocity be \( \vec{v}_g = 4 \ \mathrm{km/hr}\) going east. Let the wind's velocity be \( \vec{v}_w \). The perceived wind is coming from the north, suggesting that the apparent velocity of the wind relative to her is solely in the north direction.
02

Determine Relative Velocity of the Wind Initially

The girl's initial perceived wind direction being from the north means the relative velocity of the wind, \( \vec{v}_w - \vec{v}_g = \begin{bmatrix} 0 \ -v_{wy} \end{bmatrix} \), where \(v_{wy}\) is the initial north vector component of wind velocity. This implies \(v_{wx} = 4 \ \mathrm{km/hr}\) to negate her eastward velocity.
03

Girl Doubles Her Speed

When the girl doubles her speed, \( \vec{v}_g = 8 \ \mathrm{km/hr}\), the wind appears to come from the northwest, which means the relative velocity vector is equal in magnitude in both north and east components, indicating \( \vec{v}_w - \vec{v}_g = \begin{bmatrix} 8 \ -v_{wy} \end{bmatrix} \).
04

Solve for Wind's Velocity Components

From the initial step, \(v_{wx} = 4\). From the doubled speed scenario, substitute \( v_{wx} - 8 = -v_{wy}\). Oversimplifying gives \( 4 - 8 = -v_{wy} \) or \(v_{wy} = -4\). The wind vector is then \( \vec{v}_w = \begin{bmatrix} 4 \ -4 \end{bmatrix} \).
05

Calculate Magnitude and Direction of Wind Velocity

The magnitude of wind velocity is \( \sqrt{4^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2} \). The wind is towards the southeast, matching the vector \(\begin{bmatrix} 4 \ -4 \end{bmatrix}\). This confirms choice (a), \(4\sqrt{2}\, \mathrm{km/hr}\) towards southeast.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
Understanding vector components is key to solving problems involving relative velocity, like the one in the exercise. A vector is a quantity that has both magnitude and direction. When dealing with vectors, especially in physics, it's often useful to break them into their components.
For instance, the wind's velocity can be split into east-west and north-south components. This means we resolve the wind's velocity into
  • x-component (east-west direction)
  • y-component (north-south direction)
By analyzing these components, we can determine the vector's overall effects. In the exercise, initially, the eastward movement of the girl cancels out one wind direction component. When she doubles her speed, it affects how we perceive the wind's direction, making the understanding of vector components essential to finding the actual wind velocity.
Kinematics
Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It describes an object's motion through parameters such as displacement, velocity, and acceleration.
In our exercise, we see a practical instance of kinematics. As the girl travels eastward, her relative motion affects her perception of the wind's direction. This situation is an application of the kinematic concept of relative velocity. Here, the girl's motion alters how the wind appears to her.
By understanding kinematics, we can apply the concept of relative velocity. This helps us model systems and predict the outcome鈥攍ike determining the true velocity and direction of the wind, as required in the exercise.
Motion in Two Dimensions
Motion in two dimensions involves analyzing movement that occurs simultaneously in two different directions, usually expressed as x and y components on a graph or plane.
In the exercise, the girl experiences motion in two dimensions as both she and the wind move across the plane defined by east-west and north-south components. Understanding this helps us appreciate how her motion bothers the apparent direction of the wind, leading us to explore components in a plane.
This type of motion requires analyzing how objects move along two axes simultaneously, and is further complicated by changes in speed and direction. In our situation, the girl doubling her speed changes her trajectory significantly, influencing the vector analysis needed to find the wind's true direction. This shows the critical role of motion in two dimensions for such analyses.

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