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 based on the characteristics of uniformly accelerated motion. In uniformly accelerated motion, the position \( x \) as a function of time \( t \) can be expressed in a quadratic form, typically represented by the equation:

\[ x = ut + \frac{1}{2} a t^2 \]

where \( u \) is the initial velocity and \( a \) is the constant acceleration.

Now, let's analyze each option:

1. **Option 1: \( \beta x = \alpha t + \alpha \beta \)**

Rearranging gives:
\[ x = \frac{\alpha}{\beta} t + \beta \]
This is a linear equation in \( t \) and does not represent uniformly accelerated motion.

2. **Option 2: \( \alpha x = \beta + t \)**

Rearranging gives:
\[ x = \frac{\beta}{\alpha} + \frac{1}{\alpha} t \]
This is also a linear equation in \( t \) and does not represent uniformly accelerated motion.

3. **Option 3: \( xt = \alpha \beta \)**

Rearranging gives:
\[ x = \frac{\alpha \beta}{t} \]
This indicates an inverse relationship between \( x \) and \( t \), which does not represent uniformly accelerated motion.

4. **Option 4: \( \alpha t = \sqrt{\beta + x} \)**

Squaring both sides gives:
\[ (\alpha t)^2 = \beta + x \]
Rearranging gives:
\[ x = (\alpha t)^2 - \beta \]
This is a quadratic equation in \( t \), which indicates that \( x \) is proportional to \( t^2 \), characteristic of uniformly accelerated motion.

Based on this analysis, the equation that represents uniformly accelerated motion is:

\[ \alpha t = \sqrt{\beta + x} \]