In a projectile motion, power of the gravitational force

Options

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

Explanation

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

### Step 1: Understand the Concept of Power
Power (P) is defined as the dot product of force (F) and velocity (V):
\[ P = \mathbf{F} \cdot \mathbf{V} \]

### Step 2: Identify the Forces in Projectile Motion
In projectile motion, the only force acting on the object (after it has been projected) is the gravitational force, which acts downward. This force can be expressed as:
\[ \mathbf{F} = -mg \mathbf{j} \]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.

### Step 3: Express the Velocity Vector
The velocity vector \( \mathbf{V} \) at any instant can be expressed in terms of its components:
\[ \mathbf{V} = V_x \mathbf{i} + V_y \mathbf{j} \]
where \( V_x \) is the horizontal component and \( V_y \) is the vertical component of the velocity.

### Step 4: Calculate the Dot Product
The dot product of the gravitational force and the velocity vector is:
\[ P = \mathbf{F} \cdot \mathbf{V} = (-mg \mathbf{j}) \cdot (V_x \mathbf{i} + V_y \mathbf{j}) \]
This simplifies to:
\[ P = -mg V_y \]

### Step 5: Express \( V_y \) in Terms of Initial Conditions
The vertical component of velocity \( V_y \) can be expressed as:
\[ V_y = u_y + a_y t \]
where \( u_y = u \sin \theta \) (the initial vertical velocity) and \( a_y = -g \) (the acceleration due to gravity). Thus:
\[ V_y = u \sin \theta - gt \]

### Step 6: Substitute \( V_y \) into the Power Equation
Substituting \( V_y \) back into the power equation gives:
\[ P = -mg (u \sin \theta - gt) \]
This can be rearranged to:
\[ P = -mg u \sin \theta + mg gt \]

### Step 7: Analyze the Result
From the expression \( P = -mg u \sin \theta + mg gt \), we can see that power varies linearly with time \( t \). The term \( mg gt \) indicates that the power increases linearly over time.

### Conclusion
The power of the gravitational force during projectile motion varies linearly with time. Therefore, the correct option is that it varies linearly with time.

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