Question

A body of mass 1kg begins to move under the action of a time-dependent force $\vec F = (2t\hat i + 3{t^2}\hat j)N$ where $\hat i$ and $\hat j$ are unit vectors along $x$ and $y$ axis. What power will be developed by the force at the time $t$?
A) $(2{t^2} + 3{t^3})W$
B) $(2{t^2} + 4{t^4})W$
C) $(2{t^3} + 3{t^4})W$
D) $(2{t^3} + 3{t^5})W$

Answer 1

Hint: In these kinds of situations the first thing important is to notice the given problem statement. Especially this is a question regarding little manipulation. First, we just have to sort what we got and then we just have to interpret the missing quantities for the equation. In this question, we will get the answer simply by putting in the equation for power.

Complete step by step answer:
Given,
Mass of the given body is $1kg$
Force vector is $\vec F = (2t\hat i + 3{t^2}\hat j)N$
Firstly we have to find the acceleration of the motion,
$
  \vec a = \dfrac{{\vec F}}{m} \\
 \Rightarrow \vec a = 2t\hat i + 3{t^2}\hat j \\
 $
Now to find the velocity we have to interpret the acceleration,
\[
  \vec a = \dfrac{{d\vec v}}{{dt}} \\
 \Rightarrow \vec v = \int_0^t {\vec adt} \\
 \Rightarrow \vec v = \int_0^t {(2t\hat i + 3{t^2}\hat j)dt} \\
 \Rightarrow \vec v = {t^2}\hat i + {t^3}\hat j \\
 \]
Therefore we have all the essential values to derive power. So, the equation of power is,
\[
  P = \vec F.\vec v \\
 \Rightarrow P = (2t\hat i + 3{t^2}\hat j).({t^2}\hat i + {t^3}\hat j) \\
 \Rightarrow P = (2{t^3} + 3{t^5})W \\
 \]
Hence option (D) is correct.

Note:
The most important thing here is to notice the given values, then we can easily derivate the missing quantities. Apart from that, it is necessary to observe what process is straight and easy to find the rest values and then we can execute the formulas. In this type of question interpreting the correct formula to be used is necessary unless it will get twisted.