Relative Velocity of Helicopter as Seen by Passenger in the Train

To determine the relative velocity of the helicopter as seen by a passenger in the train, we need to consider the velocities of both the helicopter and the train and their directions of motion.

Given:
- Velocity of the helicopter = 50 km/hr (south)
- Velocity of the train = 50 km/hr (east)

Step 1: Understanding Relative Velocity
Relative velocity refers to the velocity of an object with respect to another object. In this case, we need to find the relative velocity of the helicopter as seen by a passenger in the train.

Step 2: Resolving Velocities
The velocity of the helicopter can be resolved into two components: one in the north-south direction and the other in the east-west direction. Similarly, the velocity of the train can be resolved into two components: one in the north-south direction and the other in the east-west direction.

Step 3: Analyzing the Components
Since the helicopter is flying south and the train is moving east, their velocities are perpendicular to each other. Therefore, the north-south component of the helicopter's velocity is zero, and the east-west component of the train's velocity is zero.

Step 4: Finding the Resultant Velocity
To find the relative velocity of the helicopter as seen by a passenger in the train, we need to find the resultant velocity by adding the velocity components of the helicopter and the train.

Step 5: Applying Pythagoras' Theorem
Using Pythagoras' theorem, we can find the magnitude of the resultant velocity. The magnitude is given by:
Resultant velocity = √[(north-south velocity)^2 + (east-west velocity)^2]

Step 6: Calculating the Resultant Velocity
In this case, since the north-south component of the helicopter's velocity is zero and the east-west component of the train's velocity is zero, the resultant velocity can be calculated as:
Resultant velocity = √[(0)^2 + (50)^2] = √(0 + 2500) = √2500 = 50√2 km/hr

Step 7: Determining the Direction
The direction of the resultant velocity can be determined using trigonometry. Since the helicopter is flying south and the train is moving east, the resultant velocity will be in the southeast direction.

Therefore, the relative velocity of the helicopter as seen by a passenger in the train is 50√2 km/hr towards the southeast direction.