A train is moving due east and a car is moving due north with equal speeds. A passenger in the train finds that the car is moving towards

Options

A. East-North direction
B. West-North direaction
C. South-East direction
D. None of these

Explanation

To solve the problem, we need to analyze the motion of both the train and the car from the perspective of a passenger in the train. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Motion of the Train and Car
- The train is moving due east with a velocity \( v \).
- The car is moving due north with the same velocity \( v \).

### Step 2: Establish the Reference Frame
- We will consider the ground as our reference frame.
- The velocity of the train can be represented as \( \vec{V}_{\text{train}} = v \hat{e} \) (where \( \hat{e} \) is the unit vector in the east direction).
- The velocity of the car can be represented as \( \vec{V}_{\text{car}} = v \hat{n} \) (where \( \hat{n} \) is the unit vector in the north direction).

### Step 3: Calculate the Relative Velocity of the Car with Respect to the Train
- The relative velocity of the car with respect to the train is given by:
\[ \vec{V}_{\text{car, train}} = \vec{V}_{\text{car}} - \vec{V}_{\text{train}} \]
- Substituting the values, we get:
\[ \vec{V}_{\text{car, train}} = v \hat{n} - v \hat{e} \]

### Step 4: Rearranging the Velocity Components
- We can factor out \( v \):
\[ \vec{V}_{\text{car, train}} = v (\hat{n} - \hat{e}) \]
- Here, \( \hat{n} \) represents the north direction and \( -\hat{e} \) represents the west direction (since it is opposite to east).

### Step 5: Interpret the Resultant Velocity
- The resultant vector \( \hat{n} - \hat{e} \) indicates that the car is moving in a direction that is a combination of north and west.
- To visualize this, we can draw a vector diagram where one vector points north and the other points west. The resultant will point in the northwest direction.

### Step 6: Conclusion
- Therefore, from the perspective of the passenger in the train, the car appears to be moving towards the west-northwest direction.

### Final Answer
The car is moving towards the **west-north direction**.

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