The position of a particle moving along the x-axis is given by x = a (t – 1) + b(t – 1) where a and b are constant, then :

To analyze the sentence and the equation provided about the position of a particle along the x-axis, we can start by rewriting the expression for clarity:

1. Understanding the Equation
   The equation is given as:
   $x(t) = a(t - 1) + b(t - 1)$
   This can be simplified by factoring out $(t - 1)$:
   $x(t) = (a + b)(t - 1)$
   Here, $a$ and $b$ are constants, and $x(t)$ represents the position of the particle at time $t$.

2. Interpreting the Result
   This position function indicates that the particle's position is linearly dependent on the time elapsed since $t = 1$. At $t = 1$, the position $x(1) = 0$. The rate of change of position with respect to time, or velocity, can be derived from the position function.

3. Finding Velocity
   To find the velocity $v(t)$, we differentiate $x(t)$ with respect to time $t$:
   $v(t) = \frac{dx}{dt} = a + b$
   The velocity is constant since it does not depend on $t$.

4. Final Summary
   The motion of the particle can be described as uniform rectilinear motion since the velocity remains constant over time at $a + b$.

Final Answer

• The position of the particle is given by the equation $x(t) = (a + b)(t - 1)$.
• The velocity of the particle is constant at $v = a + b$.