Question

A point object is moving with velocity ${\vec V_0} = 2\hat i - 3\hat j + 4\hat k$ in front of a moving plane mirror whose normal is along x-axis. The mirror is moving with velocity ${\vec V_m} = \hat i - 4\hat j + 3\hat k$ . Find the velocity vector of the image:
A. \[5\hat j\]
B. $ - 3\hat j + 4\hat k$
C. $ - 4\hat j + 2\hat k$
D. $2\hat i - 3\hat j + 2\hat k$

Answer 1

Hint: The pace at which an object's position changes is represented by a velocity vector. A velocity vector's magnitude indicates an object's speed. The concept of vector addition is applied in solving this problem. First, find the velocity of an object concerning the mirror and then the velocity of the image concerning the mirror. The addition of these two values will give the velocity vector of the image.


Complete answer:
Here before starting the solution firstly, we write all the given things,
Velocity of object, $\vec{V_{o}}=2\hat{i}-3\hat{j}+4\hat{k}$
Velocity of mirror, $\vec{V_{o}}=\hat{i}-4\hat{j}+\hat{k}$
Firstly, we calculate the velocity of an object w.r.t the mirror,
 ${\vec{V_{om}} = \vec {V_o} - \vec {V_m}}$
By putting all the values in the above equation, we get,
$\vec {V_{om}} = (2\hat i - 3\hat j + 4\hat k) - (\hat i - 4\hat j + 3\hat k)$
By doing further solutions in the above equation, we get the result as,
$\vec {V_{om}} = \hat i + \hat j + 2\hat k$
Here, from the above equation, we get the velocity of an object with respect to the mirror.
Now, we have to find the velocity of the image with respect to the mirror, we know the mirror is in the opposite direction of the object as from which the image velocity is quite opposite to the velocity of the object with respect to the mirror, and only changes made in X-axis and all other in the left signs,
$\vec{ V_{im}} = - \hat i + \hat j + 2\hat k$
Here, in this question, we have to find the velocity vector of the image for which we also write it as,
$\vec {V_{im}} = \vec {V_i} - \vec {V_m}$
As from the above equation, we need the velocity of the image so we take the velocity of an image on the one side and others on the next side,
$\vec {V_i} = \vec{ V_{im}} + \vec{ V_m}$
Now, we have all the values of the above equation, now we put all the values in the above equation,
$\vec {V_i} = ( - \hat i + \hat j + 2\hat k) + (\hat i - 4\hat j + 2\hat k)$
By doing the solution of the above equation, we get that,
$\vec {V_i} = - 3\hat j + 4\hat k$
Therefore, the correct answer for the velocity vector of the image is $\vec{ V_i} = - 3\hat j + 4\hat k$
The correct option is B.



Note:Make sure to know the complete concept of vector addition and subtraction before proceeding to the calculation part. A simple mistake in sign (positive or negative) can change the final answer, even though the magnitude value is correct.