Question

A particle has an initial velocity of \[3\widehat i + 4\widehat j\] and an acceleration of \[0.4\widehat i + 0.3\widehat j\]. Its speed after \[10\,{\text{s}}\] is:
A. \[10\]units
B. \[{\text{7}}\sqrt 2 \]units
C. \[{\text{7}}\]units
D. \[8.5\]units

Answer 1

Hint:We are asked to find the speed of the particle after \[10\,{\text{s}}\]. Recall the first equation of motion. The initial velocity and acceleration of the particle are given; use these values to find the final velocity of the particle. Convert the value obtained into scalar form.

Complete step by step answer:
Given, initial velocity of particle \[\overrightarrow u = 3\widehat i + 4\widehat j\]
Acceleration, \[\overrightarrow a = 0.4\widehat i + 0.3\widehat j\]
Time, \[t = 10\,{\text{s}}\]

We need to find the velocity after time, \[t = 10\,{\text{s}}\]
From first equation of motion we have,
\[v = u + at\]
In vector form, it will be,
\[\overrightarrow v = \overrightarrow u + \overrightarrow a t\]
Putting the values of \[u\], \[a\] and \[t\], we have
\[\overrightarrow v = \left( {3\widehat i + 4\widehat j} \right) + \left( {0.4\widehat i + 0.3\widehat j} \right)10 \\
\Rightarrow \overrightarrow v = 3\widehat i + 4\widehat j + 4\widehat i + 3\widehat j \\
\Rightarrow \overrightarrow v = 7\widehat i + 7\widehat j \]
We are asked to find the magnitude of final velocity.
For finding the magnitude of a vector, say \[\overrightarrow A = a\widehat i + b\widehat j + c\widehat k\], we use the formula,
\[\left| {\overrightarrow A } \right| = \sqrt {{a^2} + {b^2} + {c^2}} \]
So, using this formula for velocity vector we get,
\[\left| {\overrightarrow v } \right| = \sqrt {{7^2} + {7^2}} = 7\sqrt 2 \]
Velocity in scalar form is speed.
Therefore, the speed of the particle after \[10\,{\text{s}}\]will be \[7\sqrt 2 \] units.

Hence, the correct answer is option B.

Note:Always remember the three equations of motions. These equations are frequently used in solving problems. Remember the formula to calculate the magnitude of a vector quantity. A vector quantity has both direction and magnitude but a scalar quantity has only magnitude. Velocity is a vector quantity but velocity without direction is speed.