The relation between position (x) and time (t) are given below for a particle moving along a straight line. Which of the following equation represents uniformly accelerated motion? [where `alpha and beta` are positive constants]

Options

A. `beta x = alpha t+alpha beta`
B. `alpha x=beta+t`
C. `xt=alphabeta`
D. `alpha t=sqrt(beta+x)`

Answer

To determine which of the given equations represents uniformly accelerated motion, we need to analyze each option by differentiating the position function \( x(t) \) to find the velocity \( v(t) \) and then differentiating the velocity to find the acceleration \( a(t) \). Uniformly accelerated motion is characterized by constant acceleration.

### Step-by-Step Solution:

1. **Check Option 1:**
\[ x(t) = \frac{\alpha}{\beta} t + \alpha \]
- **Find Velocity:**
\[ v(t) = \frac{dx}{dt} = \frac{\alpha}{\beta} \]
- **Find Acceleration:**
\[ a(t) = \frac{dv}{dt} = 0 \]
- Since the acceleration is zero (not constant), this option does not represent uniformly accelerated motion.

2. **Check Option 2:**
\[ x(t) = \frac{\beta}{\alpha} + \frac{t}{\alpha} \]
- **Find Velocity:**
\[ v(t) = \frac{dx}{dt} = \frac{1}{\alpha} \]
- **Find Acceleration:**
\[ a(t) = \frac{dv}{dt} = 0 \]
- Again, the acceleration is zero (not constant), so this option also does not represent uniformly accelerated motion.

3. **Check Option 3:**
\[ x(t) = \alpha \beta t^{-1} \]
- **Find Velocity:**
\[ v(t) = \frac{dx}{dt} = -\alpha \beta t^{-2} \]
- **Find Acceleration:**
\[ a(t) = \frac{dv}{dt} = 2\alpha \beta t^{-3} \]
- The acceleration is not constant (it varies with time), so this option does not represent uniformly accelerated motion.

4. **Check Option 4:**
\[ \alpha t = \sqrt{\beta + x} \]
- **Square Both Sides:**
\[ \alpha^2 t^2 = \beta + x \implies x = \alpha^2 t^2 - \beta \]
- **Find Velocity:**
\[ v(t) = \frac{dx}{dt} = 2\alpha^2 t \]
- **Find Acceleration:**
\[ a(t) = \frac{dv}{dt} = 2\alpha^2 \]
- The acceleration is constant (\(2\alpha^2\)), indicating that this option represents uniformly accelerated motion.

### Conclusion:
The equation that represents uniformly accelerated motion is **Option 4**.