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Question: An old man and a boy are walking towards each other and a bird is flying over them as shown in the f...

An old man and a boy are walking towards each other and a bird is flying over them as shown in the figure. Find the velocity of the bird as seen by the boy.

A. 12j^12\hat{j}
B.16j^16\hat{j}
C.−12j^-12\hat{j}
D.−16j^-16\hat{j}

Explanation

Solution

From the figure we have the directions and velocities of the bird and the boy. To find the velocity of the bird as seen by the boy we need to find the relative motion of the bird with respect to the boy. For that first, we need to find the velocities of the bird and boy with respect to ground.

Complete step-by-step solution:
In the question, we have a boy and an old man walking towards each other and a bird flying over them.
In the figure, we are given the velocity vectors of the boy, the old man, and the bird.
We need to find the velocity of the bird as seen by the boy; i.e. velocity of the bird with respect to the boy.
First of all, let us find the velocity of the boy and the bird with respect to the ground.
Let ′V⃗B′'{{\vec{V}}_{B}}' be the velocity of the boy.
From the figure, it is clear that the boy is running in i^\hat{i} direction with a velocity of 16m/s.
Therefore the velocity of the boy with respect to ground,
V⃗BG=16i^{{\vec{V}}_{BG}}=16\hat{i}
Now, let ′V⃗R′'{{\vec{V}}_{R}}' be the velocity of the bird.
In the figure, it is given that the bird is flying with a velocity of 20 m/s.
Its i^\hat{i} component is given as 16m/s and j^\hat{j} component is given as 12m/s.
Therefore, the velocity of the bird with respect to the ground,
V⃗RG=(16i^+12j^){{\vec{V}}_{RG}}=\left( 16\hat{i}+12\hat{j} \right)
We have to find the velocity of the bird as seen by the boy,
V⃗RB=V⃗RG−V⃗BG V⃗RB=(16i^+12j^)−(16i^) V⃗RB=12j^ \begin{aligned} & {{{\vec{V}}}_{RB}}={{{\vec{V}}}_{RG}}-{{{\vec{V}}}_{BG}} \\\ & {{{\vec{V}}}_{RB}}=\left( 16\hat{i}+12\hat{j} \right)-\left( 16\hat{i} \right) \\\ & {{{\vec{V}}}_{RB}}=12\hat{j} \\\ \end{aligned}
Therefore, the velocity of the bird as seen by the boy is 12j^12\hat{j}

Hence the correct answer is option A.

Note: Motion of an object relative to the motion of another object in the same frame of reference is known as relative motion.
Set of coordinates that are used to determine positions and velocities of an object in the frame is called a frame of reference.
The position of a point in a plane or in a space with respect to the origin is called the position vector.