Path of the Object Dropped Through the Window of a Fast Running Train

Introduction:
When an object is dropped through the window of a fast running train, its path can be determined by considering the motion of the train and the gravitational pull on the object. Let's analyze the possible paths that the object can follow.

Options Analysis:
a) The object moves straight horizontally: This option is incorrect because the object will experience a downward force due to gravity, causing it to follow a curved path.

b) To a man standing near the track, the path of the object will be a part of a parabola: This option is correct. The path of the object can be represented by a parabolic curve due to the combination of the horizontal motion of the train and the vertical motion due to gravity.

c) The object falls down vertically: This option is incorrect because the object will not fall straight down due to the horizontal motion of the train. It will have a horizontal component of velocity as well.

d) The object will follow an elliptical path: This option is incorrect because an elliptical path requires a central force acting on the object, which is not the case here. The only force acting on the object is gravity.

Explanation:
When the object is dropped through the window of a fast running train, it will inherit the train's horizontal velocity since it was initially at rest with respect to the train. However, the object will also experience a downward force due to gravity.

The horizontal velocity of the train provides the object with a constant horizontal component of motion. Meanwhile, the vertical motion of the object is affected by gravity, causing it to accelerate downward at a rate of approximately 9.8 m/s².

As a result, the object follows a parabolic path. This is because the horizontal and vertical motions are independent of each other, and the object covers equal horizontal distances in equal time intervals, while its vertical displacement increases as the square of the time.

To a man standing near the track, the path of the object will appear as a part of a parabola, with the highest point of the curve corresponding to the point where the object was dropped from the train.

Conclusion:
When an object is dropped through the window of a fast running train, its path will be a part of a parabola. This is due to the combination of the train's horizontal motion and the gravitational pull on the object, causing it to follow a curved trajectory.

This is due to dsudo force which comes to play in non inertial frames.their are two forces also one is gravitational and another is due to motion of the train