Which of the following represents uniformly accelerated motion ?

Options

A. `x=sqrt((t+a)/(b))`
B. `x=(t+a)/(b)`
C. `t=sqrt((x+a)/(b))`
D. `x=sqrt(t+a)`

Explanation

To determine which of the given options represents uniformly accelerated motion, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Equation of Motion**:
The general equation for uniformly accelerated motion is given by:
\[ x = x_i + ut + \frac{1}{2}at^2 \]
where:
- \( x \) is the final position,
- \( x_i \) is the initial position,
- \( u \) is the initial velocity,
- \( a \) is the constant acceleration,
- \( t \) is the time.

2. **Analyze Each Option**:
We need to compare each option with the standard equation of motion to see if it matches.

- **Option A**: \( x = \sqrt{t} + \frac{a}{b} \)
- This does not have a \( t^2 \) term, hence it does not represent uniformly accelerated motion.

- **Option B**: \( x = b + a \)
- This is a constant equation with no time variable, thus it does not represent motion, let alone uniformly accelerated motion.

- **Option C**: \( t = \sqrt{x + \frac{a}{b}} \)
- Rearranging this gives us:
\[ t^2 = x + \frac{a}{b} \]
- Now, if we express \( x \):
\[ x = t^2 - \frac{a}{b} \]
- This can be rewritten as:
\[ x = -\frac{a}{b} + t^2 \]
- This matches the form of the equation of motion where \( x_i = -\frac{a}{b} \), \( u = 0 \), and \( a = b \).

- **Option D**: \( x = -b + bt^2 \)
- This can be rearranged to:
\[ x = -b + bt^2 \]
- This does not match the standard form of uniformly accelerated motion.

3. **Conclusion**:
After analyzing all options, we find that **Option C** is the only one that can be rearranged to fit the equation of uniformly accelerated motion.

### Final Answer:
**Option C** represents uniformly accelerated motion.