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A body is moving unidirectionally under the influence of a source of constant power supplying energy. Which of the diagrams shown in figure correctly shown the displacement time curve for its motion?
Options:
- A
- B
- C
- D
Solution:
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Verified Answer
The correct answer is:
As given that power $=$ constant As we know that power $(P)$
$$
P=\frac{d W}{d t}=\frac{\vec{F} \cdot \overrightarrow{d x}}{d t}=\frac{F d x}{d t}
$$
As the body is moving unidirectionally.
Hence, $F \cdot d x=F d x \cos 0^{\circ}=F d x$
$$
P=\frac{F d x}{d t}=\text { constant }
$$
( $\because P=$ constant by question)
Now, by dimensional formula
$$
\begin{aligned}
&F \cdot v=0 \\
&{[F][v]=\text { constant }} \\
&{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{LT}^{-1}\right]=\text { constant }} \\
&{\left[\mathrm{ML}^2 \mathrm{~T}^{-3}\right]=\text { constant }} \\
&\mathrm{L}^2=\frac{\mathrm{T}^3}{\mathrm{M}} \text { (As mass of body constant) } \\
&\mathrm{L}^2 \propto \mathrm{T}^3 \Rightarrow \mathrm{L} \propto \mathrm{T}^{3 / 2} \\
&\Rightarrow \text { Displacement }(d) \propto t^{3 / 2} \\
&\text { Verifies the graph (b). }
\end{aligned}
$$
$$
P=\frac{d W}{d t}=\frac{\vec{F} \cdot \overrightarrow{d x}}{d t}=\frac{F d x}{d t}
$$
As the body is moving unidirectionally.
Hence, $F \cdot d x=F d x \cos 0^{\circ}=F d x$
$$
P=\frac{F d x}{d t}=\text { constant }
$$
( $\because P=$ constant by question)
Now, by dimensional formula
$$
\begin{aligned}
&F \cdot v=0 \\
&{[F][v]=\text { constant }} \\
&{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{LT}^{-1}\right]=\text { constant }} \\
&{\left[\mathrm{ML}^2 \mathrm{~T}^{-3}\right]=\text { constant }} \\
&\mathrm{L}^2=\frac{\mathrm{T}^3}{\mathrm{M}} \text { (As mass of body constant) } \\
&\mathrm{L}^2 \propto \mathrm{T}^3 \Rightarrow \mathrm{L} \propto \mathrm{T}^{3 / 2} \\
&\Rightarrow \text { Displacement }(d) \propto t^{3 / 2} \\
&\text { Verifies the graph (b). }
\end{aligned}
$$
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