A person is travelling in a train moving due south with velocity 80 kmph his friend Q is travelling in a car moving due west with velocity 60 kmph. Q finds that his friend P is travelling with velocity

Options

A. 200 kmph `tan^(-1)` (4/3) NE
B. 100 kmph `tan^(-1)` (4/3) E of S
C. 100 kmph S - E
D. 100 kmph `tan^(-1)` (4/3) S of E

Answer

To solve the problem of finding the velocity of person P (travelling in a train) with respect to person Q (travelling in a car), we can follow these steps:

### Step 1: Understand the velocities
- Person P is travelling due south with a velocity of 80 km/h.
- Person Q is travelling due west with a velocity of 60 km/h.

### Step 2: Set up the coordinate system
- Let’s assume the positive y-axis points north and the positive x-axis points east.
- Therefore, the velocity of P can be represented as \( \vec{V_P} = (0, -80) \) km/h (since south is negative y-direction).
- The velocity of Q can be represented as \( \vec{V_Q} = (-60, 0) \) km/h (since west is negative x-direction).

### Step 3: Calculate the relative velocity
To find the velocity of P with respect to Q, we use the formula:
\[ \vec{V_{PQ}} = \vec{V_P} - \vec{V_Q} \]
Substituting the values:
\[ \vec{V_{PQ}} = (0, -80) - (-60, 0) = (0 + 60, -80) = (60, -80) \text{ km/h} \]

### Step 4: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity \( \vec{V_{PQ}} \) can be calculated using the Pythagorean theorem:
\[ |\vec{V_{PQ}}| = \sqrt{(60)^2 + (-80)^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100 \text{ km/h} \]

### Step 5: Calculate the direction of the relative velocity
To find the angle \( \theta \) that the velocity vector makes with the east direction, we use:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-80}{60} \]
Thus,
\[ \tan(\theta) = -\frac{4}{3} \]
This indicates that the angle is in the fourth quadrant (south of east). Therefore, we can find \( \theta \) using:
\[ \theta = \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 6: Conclusion
Thus, person Q finds that person P is travelling with a velocity of 100 km/h at an angle \( \tan^{-1}\left(\frac{4}{3}\right) \) south of east.

### Final Answer
The correct option is:
**100 km/h at \( \tan^{-1}\left(\frac{4}{3}\right) \) south of east.**

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