Can a body possess K.E. without having momentum?

Answer

To determine whether a body can possess kinetic energy without having momentum, let's analyze the concepts of kinetic energy and momentum step by step.

### Step 1: Understanding Kinetic Energy
Kinetic energy (K.E.) is defined as the energy that a body possesses due to its motion. The formula for kinetic energy is given by:

\[ K.E. = \frac{1}{2} mv^2 \]

where \( m \) is the mass of the body and \( v \) is its velocity.

### Step 2: Understanding Momentum
Momentum (p) is defined as the product of the mass and velocity of an object. The formula for momentum is:

\[ p = mv \]

### Step 3: Analyzing the Relationship
From the formulas, we can see that both kinetic energy and momentum depend on the mass and velocity of the body. If the body has a non-zero velocity (i.e., \( v \neq 0 \)), then:

- The kinetic energy will be non-zero because \( K.E. = \frac{1}{2} mv^2 \) will yield a positive value.
- The momentum will also be non-zero because \( p = mv \) will yield a positive value.

### Step 4: Considering Zero Velocity
If the velocity of the body is zero (i.e., \( v = 0 \)), then both kinetic energy and momentum will be zero:

- \( K.E. = \frac{1}{2} m(0)^2 = 0 \)
- \( p = m(0) = 0 \)

### Step 5: Conclusion
Since kinetic energy is dependent on the square of the velocity and momentum is directly proportional to the velocity, a body cannot possess kinetic energy without having momentum. If a body has kinetic energy, it must have a non-zero velocity, which implies that it also has momentum.

Thus, the answer to the question is **No**, a body cannot possess kinetic energy without having momentum.

---