The kinetic energy of a particle having mass m is E. Its momentum will be

Question

Accepted Answer

Here's a detailed explanation for the given question:

**Justification for Option 3 being correct:**

To find the relationship between kinetic energy (E), mass (m), and momentum (p), we start with their fundamental definitions:

1.  **Kinetic Energy (E):** The energy possessed by a body due to its motion is given by the formula:
    $E = \frac{1}{2}mv^2$
    where 'm' is the mass of the particle and 'v' is its velocity.

2.  **Momentum (p):** The product of a particle's mass and velocity is its momentum:
    $p = mv$

Now, we want to express momentum (p) in terms of mass (m) and kinetic energy (E).
From the momentum equation, we can express velocity (v) as:
$v = \frac{p}{m}$

Substitute this expression for 'v' into the kinetic energy equation:
$E = \frac{1}{2}m \left(\frac{p}{m}\right)^2$
$E = \frac{1}{2}m \frac{p^2}{m^2}$
$E = \frac{p^2}{2m}$

Now, we need to solve this equation for 'p':
Multiply both sides by $2m$:
$2mE = p^2$

Take the square root of both sides to find 'p':
$p = \sqrt{2mE}$

This derived formula directly matches Option 3.

**Explanation for why other options are incorrect:**

*   **Option 1: $2mE$**
    This option is dimensionally incorrect. The units of momentum are $kg \cdot m/s$. The units of $2mE$ would be $kg \cdot (kg \cdot (m/s)^2) = kg^2 \cdot m^2/s^2$. Since the units do not match, this option cannot be correct. Mathematically, it lacks the square root, which is essential as shown in the derivation.

*   **Option 2: $\sqrt{\frac{1}{2}mE}$**
    This option is incorrect because it has an extra factor of $\frac{1}{2}$ inside the square root. Our derivation showed that $p = \sqrt{2mE}$, not $\sqrt{\frac{1}{2}mE}$. If we square this option, we get $p^2 = \frac{1}{2}mE$, which contradicts the correct relationship $p^2 = 2mE$.

*   **Option 4: $\frac{1}{2}mE$**
    Similar to Option 1, this option is dimensionally incorrect. The units of $\frac{1}{2}mE$ are $kg^2 \cdot m^2/s^2$, which do not match the units of momentum ($kg \cdot m/s$). Also, it lacks the square root, making it mathematically incorrect.

Therefore, based on the fundamental definitions of kinetic energy and momentum, and algebraic manipulation, Option 3 is the only correct expression for momentum in terms of mass and kinetic energy.