Which of the following relations representing displacement x (t) of particle describes motion with constant acceleration?

Explanation

To determine which of the given relations for displacement \( x(t) \) describes motion with constant acceleration, we need to analyze each option by differentiating the displacement function to find the velocity and then the acceleration. Constant acceleration means that the acceleration does not depend on time.

### Step-by-Step Solution:

1. **Understand the Concept of Constant Acceleration**:
- Constant acceleration means that the acceleration \( a \) is a constant value and does not change with time \( t \) or displacement \( x \).

2. **Differentiate the Displacement Function**:
- The first step is to differentiate the displacement function \( x(t) \) to find the velocity \( v(t) \):
\[ v(t) = \frac{dx}{dt} \]

3. **Differentiate Again to Find Acceleration**:
- Next, differentiate the velocity function to find the acceleration \( a(t) \):
\[ a(t) = \frac{dv}{dt} \]

4. **Analyze Each Option**:
- **Option 1**: \( x(t) = 6 - 7t^2 \)
- Differentiate once:
\[ v(t) = \frac{d}{dt}(6 - 7t^2) = -14t \]
- Differentiate again:
\[ a(t) = \frac{d}{dt}(-14t) = -14 \]
- Since \( a(t) = -14 \) is a constant, this option represents motion with constant acceleration.

- **Option 2**: \( x(t) = 6 + 7t^3 \)
- Differentiate once:
\[ v(t) = \frac{d}{dt}(6 + 7t^3) = 21t^2 \]
- Differentiate again:
\[ a(t) = \frac{d}{dt}(21t^2) = 42t \]
- Since \( a(t) = 42t \) depends on \( t \), this option does not represent constant acceleration.

- **Option 3**: \( x(t) = 6 + \frac{7}{t} \)
- Differentiate once:
\[ v(t) = \frac{d}{dt}(6 + \frac{7}{t}) = -\frac{7}{t^2} \]
- Differentiate again:
\[ a(t) = \frac{d}{dt}(-\frac{7}{t^2}) = \frac{14}{t^3} \]
- Since \( a(t) = \frac{14}{t^3} \) depends on \( t \), this option does not represent constant acceleration.

- **Option 4**: \( x(t) = 6 + 7\sqrt{t} \)
- Differentiate once:
\[ v(t) = \frac{d}{dt}(6 + 7\sqrt{t}) = \frac{7}{2\sqrt{t}} \]
- Differentiate again:
\[ a(t) = \frac{d}{dt}(\frac{7}{2\sqrt{t}}) = -\frac{7}{4t^{3/2}} \]
- Since \( a(t) = -\frac{7}{4t^{3/2}} \) depends on \( t \), this option does not represent constant acceleration.

5. **Conclusion**:
- The only option that describes motion with constant acceleration is **Option 1**: \( x(t) = 6 - 7t^2 \).