In a projectile motion, power of the gravitational force

Options

A. is constant throughout
B. is negative for first half, zero at topmost point and positive for rest half
C. varies linearly with time
D. is positive for complete path

Answer

To find the power of the gravitational force during projectile motion, we can follow these steps:

### Step 1: Understand the Concept of Power
Power is defined as the rate at which work is done or energy is transferred. In the context of forces, power can be calculated using the formula:
\[ P = F \cdot v \]
where \( P \) is power, \( F \) is the force acting on the object, and \( v \) is the velocity of the object in the direction of the force.

### Step 2: Identify the Forces Acting on the Projectile
In projectile motion, the only force acting on the projectile (after it has been launched) is the gravitational force, which can be expressed as:
\[ F = mg \]
where \( m \) is the mass of the projectile and \( g \) is the acceleration due to gravity.

### Step 3: Determine the Velocity of the Projectile
The velocity of the projectile can be broken down into its components. The vertical component of the initial velocity can be expressed as:
\[ u_y = u \sin \theta \]
where \( u \) is the initial velocity and \( \theta \) is the angle of projection. As the projectile moves upwards, its vertical velocity changes due to the effect of gravity.

### Step 4: Write the Expression for Power
The power of the gravitational force can be expressed as:
\[ P = F \cdot v_y \]
Substituting the values of force and vertical velocity:
\[ P = mg \cdot (u_y + gt) \]
This equation shows that the power depends on both the gravitational force and the vertical velocity of the projectile.

### Step 5: Analyze the Power During Different Phases of Motion
- **Ascent Phase**: When the projectile is moving upwards, the vertical velocity \( v_y \) decreases until it reaches zero at the highest point. Therefore, the power will be positive but decreasing.
- **At the Highest Point**: The vertical velocity becomes zero, and hence the power becomes zero.
- **Descent Phase**: When the projectile starts descending, the vertical velocity increases, and the power becomes positive again.

### Conclusion
The power of the gravitational force changes throughout the projectile's motion. It is positive during the ascent (but decreasing), zero at the highest point, and positive during the descent (increasing).

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