A missile is launched with a velocity less than the escape velocity. The sum of its kinetic and potential energy is

Options

A. Positive
B. Negative
C. May be negative or positive depending upon its initial velocity
D. Zero

Answer

To solve the problem, we need to analyze the situation where a missile is launched with a velocity less than the escape velocity. We will consider the kinetic energy (KE) and potential energy (PE) of the missile.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
- The escape velocity (\(V_{esc}\)) is the minimum velocity required for an object to break free from the gravitational attraction of a celestial body (like Earth) without any additional propulsion.
- If the missile is launched with a velocity less than the escape velocity, it means it does not have enough energy to escape the gravitational field of the Earth.

2. **Kinetic and Potential Energy**:
- The kinetic energy (\(KE\)) of the missile is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \(m\) is the mass of the missile and \(v\) is its velocity.
- The gravitational potential energy (\(PE\)) at a distance \(r\) from the center of the Earth is given by:
\[ PE = -\frac{GMm}{r} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the distance from the center of the Earth.

3. **Total Mechanical Energy**:
- The total mechanical energy (\(E\)) of the missile is the sum of its kinetic and potential energy:
\[ E = KE + PE \]

4. **Analyzing the Situation**:
- Since the missile's velocity is less than the escape velocity, it will not have enough kinetic energy to overcome the gravitational potential energy.
- As the missile rises, it will lose kinetic energy and gain potential energy. However, the total energy will remain negative because the kinetic energy is not sufficient to make the total energy zero or positive.

5. **Conclusion**:
- Therefore, when the missile is launched with a velocity less than the escape velocity, the sum of its kinetic and potential energy will be negative.

### Final Answer:
The sum of the kinetic and potential energy of the missile is **negative**.

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