A particle is projected at `t=0` from a point on the ground with certain velocity at an angle with the horizontal. The power of gravitation force is plotted against time. Which of the following is the best representation?

Options

A.
B.
C.
D.

Explanation

To solve the problem step by step, we need to analyze the power of the gravitational force acting on a particle projected at an angle. Here’s how we can break it down:

### Step 1: Understand the scenario
A particle is projected from the ground at an angle θ with an initial velocity u. The motion of the particle will follow a projectile path under the influence of gravity.

### Step 2: Resolve the initial velocity
The initial velocity can be resolved into two components:
- Horizontal component: \( u_x = u \cos \theta \)
- Vertical component: \( u_y = u \sin \theta \)

### Step 3: Identify the force acting on the particle
The only force acting on the particle after projection is the gravitational force, which is directed downward:
- Gravitational force, \( F_g = -mg \hat{j} \)

### Step 4: Write the velocity vector at any time t
At any time t, the velocity vector \( \vec{v} \) can be expressed as:
\[ \vec{v} = u \cos \theta \hat{i} + (u \sin \theta - gt) \hat{j} \]
Here, \( -gt \) accounts for the downward acceleration due to gravity.

### Step 5: Calculate the power at any instant
Power \( P \) is defined as the dot product of the force and velocity:
\[ P = \vec{F} \cdot \vec{v} \]
Substituting the values:
\[ P = (-mg \hat{j}) \cdot (u \cos \theta \hat{i} + (u \sin \theta - gt) \hat{j}) \]
This simplifies to:
\[ P = -mg (u \sin \theta - gt) \]
Thus, we can express the power as:
\[ P = -mg u \sin \theta + mg gt \]

### Step 6: Analyze the power with respect to time
From the equation \( P = -mg u \sin \theta + mg gt \), we see that:
- The term \( mg gt \) indicates that power increases linearly with time.
- The initial power will be negative (when the particle is moving upwards against gravity) and will become positive after reaching the peak (when the particle starts moving downwards).

### Step 7: Conclusion
The power of the gravitational force plotted against time will show a linear relationship. Initially, the power will be negative and will increase to positive as time progresses.

### Final Representation
The best representation of the power of gravitational force against time is a linear graph that starts below the time axis (negative power) and crosses to above the time axis (positive power).