A projectile is launched from the surface of earth with a velocity less than the escape velocity. Its total mechanical energy is

Options

A. positive
B. negative
C. zero
D. dependes upon initial velocity

Explanation

To determine the total mechanical energy of a projectile launched from the surface of the Earth with a velocity less than the escape velocity, we can follow these steps:

### Step 1: Understand Escape Velocity
The escape velocity (\(v_e\)) from the surface of the Earth is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the Earth. For our case, we know that the projectile is launched with a velocity \(v_0\) that is less than \(v_e\).

### Step 2: Calculate Potential Energy
The gravitational potential energy (\(U\)) of the projectile at the surface of the Earth is given by:
\[ U = -\frac{GMm_0}{R} \]
where \(m_0\) is the mass of the projectile. The negative sign indicates that the gravitational force is attractive.

### Step 3: Calculate Kinetic Energy
The kinetic energy (\(K\)) of the projectile when it is launched with velocity \(v_0\) is:
\[ K = \frac{1}{2} m_0 v_0^2 \]

### Step 4: Total Mechanical Energy
The total mechanical energy (\(E\)) of the projectile is the sum of its potential energy and kinetic energy:
\[ E = U + K \]
Substituting the expressions for \(U\) and \(K\):
\[ E = -\frac{GMm_0}{R} + \frac{1}{2} m_0 v_0^2 \]

### Step 5: Analyze the Condition of \(v_0 < v_e\)
Since \(v_0\) is less than the escape velocity, we can analyze the scenario when \(v_0\) is equal to \(v_e\):
\[ v_e = \sqrt{\frac{2GM}{R}} \implies \frac{1}{2} v_e^2 = \frac{GM}{R} \]
Thus, if \(v_0 = v_e\), the total energy would be:
\[ E = -\frac{GMm_0}{R} + \frac{GMm_0}{R} = 0 \]
Since \(v_0 < v_e\), the kinetic energy will be less than \(\frac{GMm_0}{R}\), leading to:
\[ E < 0 \]

### Conclusion
Therefore, the total mechanical energy of the projectile launched with a velocity less than the escape velocity is negative.

### Final Answer
The total mechanical energy is negative.

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