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The potential energy of a particle having mass $m$ is given by $U = \dfrac{1}{2}k{x^2}$ for $x < 0$ and $U = 0$ for $x \geqslant 0$. If the total mechanical energy of the particle is $E$, then its speed at $x = \sqrt {\dfrac{{2E}}{k}} $ is
(A) Zero
(B) $\sqrt {\dfrac{{2E}}{m}} $
(C) $\sqrt {\dfrac{{2E}}{m}} $
(D) $\sqrt {\dfrac{E}{{2m}}} $

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Last updated date: 13th Jun 2024
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Answer
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Hint Here, write the potential energy and substitute the value of distance at which speed is to be determined. Then, substitute the value of energy to the mechanical energy expression to find kinetic energy and then in the expression of kinetic energy to find the speed.
Formula Used: Here we will be using the formula for conservation of mechanical energy is $E = U + K$, where $E$ is the mechanical energy, $U$ is the potential energy and $K$ is the kinetic energy.

Complete step by step solution
The given potential energy of a particle of mass $m$ for $x < 0$ is $U = \dfrac{1}{2}k{x^2}$, here, the distance travelled in $x$ axis and the potential energy for $x \geqslant 0$ is $U = 0$.
The total mechanical energy of the particle is $E$.
The value of distance at which speed is to be determined is,
$ \Rightarrow x = \sqrt {\dfrac{{2E}}{k}} $
Substitute $\sqrt {\dfrac{{2E}}{k}} $ for $x$ in the expression of potential energy for $x < 0$.
$ \Rightarrow U = \dfrac{1}{2}k{\left( {\sqrt {\dfrac{{2E}}{k}} } \right)^2}$
Reduce the above equation.
$ \Rightarrow U = \dfrac{1}{2}k \times \dfrac{{2E}}{k}$
Cancel out the same terms in division.
$ \Rightarrow U = E$
According to conservation of energy, mechanical energy is expressed as:
$E = U + K$
From the above, substitute $E$ for $U$ in the above expression.
$E = E + K$
Take the same variable to one side to determine the value of kinetic energy and rearrange the expression.
$K = E - E = 0$
Formula for kinetic energy is given by
$K = \dfrac{1}{2}m{v^2}$
Here, $m$ is the mass and $v$ is the velocity or speed measured in meters per second.
Substitute $0$ for $K$ in the above expression.
$ \Rightarrow 0 = \dfrac{1}{2}m{v^2}$
$ \Rightarrow v = 0$

So, option (B) is correct answer

Additional information Energy possessed by an object due to its motion or position is defined as the mechanical energy. According to physical science mechanical energy is also defined as the sum of kinetic energy and potential energy.

Note Principle of conservation of mechanical energy is known to solve the questions. Also, for stationary position kinetic energy equals to zero. Thus, kinetic energy is to be only determined when the object is in moving condition.