A particle of mass an charge `q` is projected vertically upwards .A uniform electric field `vec(E)` is acted vertically downwards.The most appropriate graph between potential energy `U` (gravitation plus electrostatic) and height h(`lt lt` radius of earth) is :(assume `U` to be zero on surface of earth)

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Answer

To solve the problem, we need to analyze the potential energy of a particle of mass \( m \) and charge \( q \) projected vertically upwards in the presence of a uniform electric field \( \vec{E} \) acting downwards. We will consider both gravitational potential energy and electrostatic potential energy.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle**:
- The gravitational force acting on the particle is given by \( F_g = mg \), acting downwards.
- The electrostatic force due to the electric field is given by \( F_e = qE \), also acting downwards since the electric field is directed downwards.

2. **Calculate the Gravitational Potential Energy**:
- The gravitational potential energy \( U_g \) at a height \( h \) is given by:
\[ U_g = mgh \]
- This potential energy is zero at the surface of the Earth (when \( h = 0 \)).

3. **Calculate the Electrostatic Potential Energy**:
- The electrostatic potential energy \( U_e \) in a uniform electric field is given by:
\[ U_e = qEh \]
- This potential energy is also considered to be zero at the surface of the Earth (when \( h = 0 \)).

4. **Total Potential Energy**:
- The total potential energy \( U \) of the particle at height \( h \) is the sum of the gravitational potential energy and the electrostatic potential energy:
\[ U = U_g + U_e = mgh + qEh \]
- Therefore, we can express the total potential energy as:
\[ U = (mg + qE)h \]

5. **Analyze the Relationship Between Potential Energy and Height**:
- From the equation \( U = (mg + qE)h \), we see that potential energy \( U \) is directly proportional to height \( h \).
- This means as height \( h \) increases, the total potential energy \( U \) also increases linearly.

6. **Graphical Representation**:
- Since \( U \) is directly proportional to \( h \), the graph of potential energy \( U \) versus height \( h \) will be a straight line starting from the origin (0,0) and increasing linearly.

### Conclusion:
The most appropriate graph between potential energy \( U \) (gravitational plus electrostatic) and height \( h \) is a straight line that passes through the origin.