A stone is projected at time `t=0` with a speed `v_(0)` at an angle `theta` with the horizontal in a uniform gravitational field. The rate of work done (P) by the gravitational force plotted against time (t) will be as.

Options

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B.
C.
D.

Answer

To solve the problem, we need to analyze the work done by the gravitational force on the stone projected at an angle. The work done by a force is given by the formula:

\[ P = \mathbf{F} \cdot \mathbf{v} \]

where \( P \) is the power (rate of work done), \( \mathbf{F} \) is the force, and \( \mathbf{v} \) is the velocity of the object.

### Step-by-Step Solution:

1. **Identify the Forces**:
The gravitational force acting on the stone is given by:
\[ \mathbf{F} = -mg \hat{j} \]
where \( m \) is the mass of the stone, \( g \) is the acceleration due to gravity, and \( \hat{j} \) is the unit vector in the vertical direction (downward).

2. **Determine the Velocity Components**:
The stone is projected with an initial speed \( v_0 \) at an angle \( \theta \). The velocity components can be expressed as:
\[ V_x = v_0 \cos(\theta) \quad \text{and} \quad V_y = v_0 \sin(\theta) - gt \]
Here, \( V_y \) decreases over time due to the gravitational acceleration \( g \).

3. **Calculate the Dot Product**:
The power can be expressed as:
\[ P = \mathbf{F} \cdot \mathbf{v} = (-mg \hat{j}) \cdot (V_x \hat{i} + V_y \hat{j}) = -mg V_y \]
Substituting for \( V_y \):
\[ P = -mg \left(v_0 \sin(\theta) - gt\right) \]
This simplifies to:
\[ P = -mg v_0 \sin(\theta) + mg^2 t \]

4. **Analyze the Expression**:
The expression for power \( P \) can be rearranged as:
\[ P = mg^2 t - mg v_0 \sin(\theta) \]
This is a linear equation in terms of time \( t \) with:
- A slope of \( mg^2 \)
- A y-intercept of \( -mg v_0 \sin(\theta) \)

5. **Conclusion**:
Since the equation is linear, the graph of power \( P \) against time \( t \) will be a straight line with a positive slope, indicating that the rate of work done by the gravitational force increases linearly with time.

### Final Answer:
The rate of work done (P) by the gravitational force plotted against time (t) will be a straight line with a positive slope.