Which of the following statements about kinetic energy (K.E.) is true?

Options

A. All objects moving with the same velocity have the same K.E.
B. The K.E. of a body will quadruple if its velocity doubles
C. As the velocity of a body increases, its K.E. decreases
D. The K.E. of a body is independent of its mass

Explanation

To solve the question regarding which statement about kinetic energy (K.E.) is true, we will analyze each statement based on the formula for kinetic energy, which is given by:

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

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

### Step-by-Step Solution:

1. **Evaluate the First Statement:**
- **Statement:** All objects moving with the same velocity have the same kinetic energy.
- **Analysis:** The kinetic energy depends on both mass and velocity. If two objects have the same velocity but different masses, their kinetic energies will be different.
- **Conclusion:** This statement is **incorrect**.

2. **Evaluate the Second Statement:**
- **Statement:** The kinetic energy of the body will quadruple if its velocity doubles.
- **Analysis:** If the velocity doubles, we can express this mathematically:
- Let the initial velocity be \( v \) and the new velocity be \( 2v \).
- Initial K.E. \( (K.E.1) = \frac{1}{2} mv^2 \)
- New K.E. \( (K.E.2) = \frac{1}{2} m(2v)^2 = \frac{1}{2} m(4v^2) = 2mv^2 \)
- Therefore, \( K.E.2 = 4 \times K.E.1 \).
- **Conclusion:** This statement is **correct**.

3. **Evaluate the Third Statement:**
- **Statement:** As the velocity of the body increases, its kinetic energy decreases.
- **Analysis:** From the formula, we see that kinetic energy is directly proportional to the square of the velocity. If the velocity increases, the kinetic energy also increases.
- **Conclusion:** This statement is **incorrect**.

4. **Evaluate the Fourth Statement:**
- **Statement:** The kinetic energy of the body is independent of mass.
- **Analysis:** The kinetic energy depends on both mass and velocity. If the mass changes while keeping the velocity constant, the kinetic energy will also change.
- **Conclusion:** This statement is **incorrect**.

### Final Answer:
The only true statement is the **second statement**: "The kinetic energy of the body will quadruple if its velocity doubles."

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