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

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

\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.