Question

A body starts with a velocity $2\hat i + 3\hat j + 11\hat k\,m{s^{ - 1}}$ and moves with an acceleration of $5\hat i + 5\hat j - 5\hat k\,m{s^{ - 2}}$ , then its velocity after $0.2\,s$ will be:

Answer 1

Hint: Use the equation of the motion formula and substitute the initial velocity, acceleration and the time taken in it to find the final velocity of the body. From the vector form of the answer calculate the magnitude of the velocity obtained by the modulus of it.
Useful formula:
The equation of the motion is given by
$v = u + at$
Where $v$ is the final velocity of the body, $u$ is the initial velocity of the body, $a$ is the acceleration of the body and the $t$ is the time taken for the change in the velocity.

Complete step by step solution:
It is given that the
Initial velocity of the body, $\vec u = 2\hat i + 3\hat j + 11\hat k\,m{s^{ - 1}}$
Acceleration of the body, $\vec a = 5\hat i + 5\hat j - 5\hat k\,m{s^{ - 2}}$
Time taken for the change in the velocity, $t = 0.2\,s$
Let us use the equation of the motion to find the final velocity of the body.
$v = u + at$
Substituting the known values in it,
\[v = 2\hat i + 3\hat j + 11\hat k + \left( {5\hat i + 5\hat j - 5\hat k} \right) \times 0.2\]
My multiplying the time taken with the acceleration, we get
\[v = 2\hat i + 3\hat j + 11\hat k + \left( {\hat i + \hat j - \hat k} \right)\,\]
By adding the similar terms in the above step, we get the following result.
$\vec v = 3\hat i + 4\hat j + 10\hat k\,m{s^{ - 1}}$
In order to find the magnitude of the velocity, modulus of the vector form of velocity is taken.
$v = \sqrt {{3^2} + {4^2} + {{10}^2}} $
By simplifying the above equation, we get
$v = \sqrt {125} \,m{s^{ - 1}}$
Hence the final velocity of the body after the time taken of $0.2\,s$ is obtained as $\sqrt {125} \,m{s^{ - 1}}$ .

Note: In the above solution, the modulus of the vector is taken. It means the square root is taken as the whole outside the vector and the individual terms are separately squared inside to find the modulus of the vector. This modulus provides the magnitude of the value.