A railway train, moving at the rate of 44 m/sec, is struck by a stone, moving horizontally and at right angles tot he train with velocity of 33 m/sec. The magnitude and direction of the velocity with which the stone appears to meet the train is

Options

A. `50 tan ^(-1) 3//4`
B. `55 tan ^(-1) (-(3)/(4))`
C. `40 cos ^(-1) "" (3)/(4)`
D. none

Answer

To solve the problem of finding the magnitude and direction of the velocity with which the stone appears to meet the train, we can follow these steps:

### Step 1: Understand the velocities involved
- The train is moving with a velocity \( V_t = 44 \, \text{m/s} \) in one direction (let's assume along the x-axis).
- The stone is moving with a velocity \( V_s = 33 \, \text{m/s} \) in the perpendicular direction (let's assume along the y-axis).

### Step 2: Represent the velocities as vectors
- The velocity of the train can be represented as a vector:
\[ \vec{V_t} = 44 \hat{i} \, \text{m/s} \]
- The velocity of the stone can be represented as a vector:
\[ \vec{V_s} = 33 \hat{j} \, \text{m/s} \]

### Step 3: Calculate the relative velocity of the stone with respect to the train
The relative velocity of the stone with respect to the train, \( \vec{V_{st}} \), is given by:
\[ \vec{V_{st}} = \vec{V_s} - \vec{V_t} \]
Substituting the values:
\[ \vec{V_{st}} = 33 \hat{j} - 44 \hat{i} \]
This can be written as:
\[ \vec{V_{st}} = -44 \hat{i} + 33 \hat{j} \]

### Step 4: Find the magnitude of the relative velocity
The magnitude of the relative velocity \( |\vec{V_{st}}| \) can be calculated using the Pythagorean theorem:
\[ |\vec{V_{st}}| = \sqrt{(-44)^2 + (33)^2} \]
Calculating the squares:
\[ = \sqrt{1936 + 1089} = \sqrt{3025} \]
Thus,
\[ |\vec{V_{st}}| = 55 \, \text{m/s} \]

### Step 5: Find the direction of the relative velocity
The direction can be found using the tangent function:
\[ \tan(\theta) = \frac{V_{sy}}{V_{tx}} = \frac{33}{44} \]
Calculating the angle \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{33}{44}\right) \]
Using a calculator,
\[ \theta \approx 39.76^\circ \]

### Conclusion
The magnitude of the velocity with which the stone appears to meet the train is \( 55 \, \text{m/s} \), and the direction is approximately \( 39.76^\circ \) above the negative x-axis (or towards the train).