What is the relation between E, P, and E if E is kinetic energy, P is momentum, and V is the velocity of the particle

The correct option is P = (dE/dV).

Let us understand the step-by-step derivation of the relation between the kinetic energy, momentum, and velocity of a particle.

Step 1: Define Kinetic Energy and Momentum
For a particle of mass m moving with a velocity V:
The kinetic energy (E) of the particle is defined by the formula:
$$E=\frac{1}{2}m{V}^2$$
The momentum (P) of the particle is defined as:
$$P=mV$$

Step 2: Differentiate Kinetic Energy with respect to Velocity
Now, let us find the derivative of the kinetic energy (E) with respect to the velocity (V):
$$\frac{dE}{dV}=\frac{d}{dV}\left(\frac{1}{2}m{V}^2\right)$$

Applying the power rule of differentiation:
$$\frac{dE}{dV}=\frac{1}{2}m·2V$$
Simplifying the expression:
$$\frac{dE}{dV}=mV$$

Step 3: Relate the derivative to Momentum
Since we know that the momentum of the particle is P = m V, we can substitute P into our simplified derivative equation:
$$\frac{dE}{dV}=P$$
Or, rewriting the relation:
$$P=\frac{dE}{dV}$$

This confirms that the momentum of a particle is equal to the derivative of its kinetic energy with respect to its velocity. Therefore, the correct relation is P = (dE/dV).