Q.A bird holds a fruit in its beak and flies parallel to the ground. It lets go of the fruit at some height. Describe the trajectory of the fruit as it falls to the ground as seen by
Concept understanding — Relative Velocity in Two Dimensions
Relative Velocity in Two Dimensions
You already know relative velocity in one dimension — the speed of a car measured from another moving car. Two dimensions is the same idea, but now velocities have both magnitude and direction, so subtraction becomes vector subtraction.
The Intuition
Imagine you're standing on a train platform. A train passes you at 15 m/s. From your frame, the train's velocity is 15 m/s forward. Now imagine you're on that train, walking toward the front at 2 m/s relative to the train. Your velocity relative to the platform is 15 + 2 = 17 m/s forward. That's one-dimensional relative velocity: just add or subtract numbers.
Now picture a boat crossing a river. The boat points straight across, but the river flows sideways. From the bank, the boat moves diagonally — its velocity is the vector sum of its own velocity (relative to water) and the water's velocity (relative to bank). To find how the boat moves relative to the bank, you add vectors.
The reverse problem: you know the boat's velocity relative to bank and the water's velocity. To find the boat's velocity relative to water, you subtract vectors. That subtraction is the core of relative velocity in two dimensions.
The Precise Statement
If object A has velocity vA and object B has velocity vB, both measured in the same frame (say, the ground), then the velocity of A relative to B is:
vA/B=vA−vB
This is a vector equation. It says: to find how A appears to move from B's perspective, take A's velocity and subtract B's velocity (vector subtraction). Equivalently, vA/B is the velocity you would measure if you were riding on B.
Important
The order matters: vA/B means "velocity of A as seen from B". Always write the first object minus the second.
Why Vector Subtraction?
In one dimension, subtraction is just flipping a sign. In two dimensions, velocities have components. Subtracting vectors means subtracting corresponding components:
If vA=vAxi^+vAyj^ and vB=vBxi^+vByj^, then
vA/B=(vAx−vBx)i^+(vAy−vBy)j^
Geometrically, you place the tails of vA and vB together. The vector from the tip of vB to the tip of vA is vA/B.
The Two Classic Problems
Rain-Man Problem: A man walks in rain. Rain falls vertically downward at speed vr relative to ground. The man walks horizontally at speed vm. To him, the rain appears to come at an angle. Why? Because the rain's velocity relative to the man is vr/m=vr−vm. Since vr is downward and vm is horizontal, the relative velocity is diagonal — the man must tilt his umbrella forward. …
At the instant of release, the fruit shares the bird's own horizontal velocity, so whether its path looks straight or curved depends entirely on which observer's frame — the bird's moving frame or the ground's fixed frame — is used to describe it. …
The two observers are in relative motion, so they perceive different trajectories for the same falling fruit — a straight vertical drop for the bird, and a curved parabolic path for the person on the ground.
At the instant the fruit is released, it has the same horizontal velocity as the bird (since it was being carried along with the bird), and from then on it falls freely under gravity (assuming negligible air resistance and that the bird continues to fly at the same constant horizontal velocity).
(a) As seen by the bird: Since the bird continues to move horizontally with the same velocity that the fruit had at release, the bird and the fruit always have the same horizontal velocity relative to each other. So, relative to the bird, the fruit has no horizontal motion — it only appears to accelerate downward due to gravity. The bird therefore sees the fruit fall in a straight vertical line, directly below the point of release (from the bird's own moving frame).
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2022Set ANNUAL4 marks
Q.A bird holds a fruit in its beak and flies parallel to the ground. It lets go of the fruit at some height. Describe the trajectory of the fruit as it falls to the ground as seen by
(a) the bird
(b) a person on the ground.
›Reveal solutionSolution
The two observers are in relative motion, so they perceive different trajectories for the same falling fruit — a straight vertical drop for the bird, and a curved parabolic path for the person on the ground.
At the instant the fruit is released, it has the same horizontal velocity as the bird (since it was being carried along with the bird), and from then on it falls freely under gravity (assuming negligible air resistance and that the bird continues to fly at the same constant horizontal velocity).
(a) As seen by the bird: Since the bird continues to move horizontally with the same velocity that the fruit had at release, the bird and the fruit always have the same horizontal velocity relative to each other. So, relative to the bird, the fruit has no horizontal motion — it only appears to accelerate downward due to gravity. The bird therefore sees the fruit fall in a straight vertical line, directly below the point of release (from the bird's own moving frame).