Question

A bird moves from point (1, -2) to (4, 2). If the speed of the bird is \[10{m}/{s}\;\], then the velocity vector of the bird is:

Answer 1

Hint: Using the formula for calculating the unit vector should be used. Then, the unit displacement vector should be calculated and the magnitude of the given velocity should be multiplied to obtain the value of the velocity vector.

Formula used:
\[\overset{\hat{\ }}{\mathop{A}}\,=\dfrac{\overset{\to }{\mathop{A}}\,}{|\overset{\to }{\mathop{A}}\,|}\]
\[\overset{\to }{\mathop{d}}\,=\overset{\to }{\mathop{{{r}_{f}}}}\,-\overset{\to }{\mathop{{{r}_{i}}}}\,\]

Complete step by step answer:
From given, we have the data,
The distance covered by a bird is from point (1, -2) to (4, 2).
\[\Rightarrow \] The initial position of the vector of the bird is, \[(1,-2)\to {{r}_{i}}=i-2j\]
The final position of the vector of the bird is, \[(4,2)\to {{r}_{f}}=4i+2j\]
The speed of the bird is \[10{m}/{s}\;\]

The displacement vector is the difference between the final and initial position of the bird.
Let us compute the displacement vector.
\[\overset{\to }{\mathop{d}}\,=\overset{\to }{\mathop{{{r}_{f}}}}\,-\overset{\to }{\mathop{{{r}_{i}}}}\,\]
Where \[\overset{\to }{\mathop{{{r}_{f}}}}\,\]is the final position and \[\overset{\to }{\mathop{{{r}_{i}}}}\,\]is the initial position.
Let us substitute the given values in the above equation.
So, we get,
\[\begin{align}
  & \overset{\to }{\mathop{d}}\,=(4i+2j)-(i-2j) \\
 & \Rightarrow \overset{\to }{\mathop{d}}\,=3i+4j \\
\end{align}\]
We have obtained the displacement vector. So, we need to find the value of the unit displacement vector.
The unit displacement vector is given as follows.
\[\overset{\hat{\ }}{\mathop{d}}\,=\dfrac{\overset{\to }{\mathop{d}}\,}{|\overset{\to }{\mathop{d}}\,|}\]
Where \[\overset{\to }{\mathop{d}}\,\]is the displacement vector and \[|\overset{\to }{\mathop{d}}\,|\]is the magnitude of the displacement vector.
Now substitute the values in the above equation.
So, we get,
\[\begin{align}
  & \overset{\hat{\ }}{\mathop{d}}\,=\dfrac{3i+4j}{|3i+4j|} \\
 & \Rightarrow \overset{\hat{\ }}{\mathop{d}}\,=\dfrac{3i+4j}{\sqrt{{{3}^{2}}+{{4}^{2}}}} \\
\end{align}\]
Upon further calculating, we get the unit vector of the displacement as follows,
\[\overset{\hat{\ }}{\mathop{d}}\,=\dfrac{3i+4j}{5}\]

The direction of both, the velocity and the displacement are the same. And the magnitude of the velocity is given to be \[10{m}/{s}\;\].
Hence the velocity vector is calculated as follows.
\[\begin{align}
  & \overset{\to }{\mathop{v}}\,=10\times \dfrac{3i+4j}{5} \\
 & \Rightarrow \overset{\to }{\mathop{v}}\,=(6i+8j)\,{m}/{s}\; \\
\end{align}\]

Therefore, the value of the velocity vector is\[(6i+8j)\,{m}/{s}\;\].

Note:
The things to be on your finger-tips for further information on solving these types of problems are: The conversion of the displacement vector to the unit displacement vector should be taken care of, otherwise the whole calculation results in the wrong final answer.