Which of the following equation represents the motion of a body moving with constant finite acceleration ? In these equation, y denotes the displacement in time t and p.q and r the arbitary constants ?

Options

A. `y=(p+qt)(r+pt)`
B. `y=p+tqr`
C. `y=(p+t)(q+t)(r+1)`
D. `y=(p+qt)r`

Explanation

To determine which of the given equations represents the motion of a body moving with constant finite acceleration, we need to analyze each equation. We will differentiate the equations to find the acceleration and check if it is constant and finite.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find which equation represents motion with constant finite acceleration. The acceleration can be determined by taking the second derivative of the displacement \( y \) with respect to time \( t \).

2. **Analyzing the First Option**:
- Equation: \( y = p + qt + rt^2 \)
- Differentiate once to find velocity:
\[ \frac{dy}{dt} = q + 2rt \]
- Differentiate again to find acceleration:
\[ \frac{d^2y}{dt^2} = 2r \]
- Since \( r \) is a constant, the acceleration is constant and finite.

3. **Analyzing the Second Option**:
- Equation: \( y = p + qt \)
- Differentiate once:
\[ \frac{dy}{dt} = q \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = 0 \]
- The acceleration is zero, which is not finite acceleration.

4. **Analyzing the Third Option**:
- Equation: \( y = p + qt + rt \)
- Differentiate once:
\[ \frac{dy}{dt} = q + r \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = 0 \]
- The acceleration is zero, which is not finite acceleration.

5. **Analyzing the Fourth Option**:
- Equation: \( y = p + qt + rt^2 \)
- Differentiate once:
\[ \frac{dy}{dt} = q + 2rt \]
- Differentiate again:
\[ \frac{d^2y}{dt^2} = 2r \]
- Since \( r \) is a constant, the acceleration is constant and finite.

### Conclusion:
From the analysis, the equations that represent motion with constant finite acceleration are:
- **First Option**: \( y = p + qt + rt^2 \)
- **Fourth Option**: \( y = p + qt + rt^2 \)

### Final Answer:
The correct equations representing the motion of a body with constant finite acceleration are the **First Option** and the **Fourth Option**.

---