The problem you've provided describes a particle moving with uniform acceleration from rest. Let's break it down step-by-step:

Given:

1. The particle starts from rest at point $O$.
2. It passes through points $P$, $Q$, and $R$.
3. Distances:
   • $|PQ| = 105 \, \text{m}$
   • $|QR| = 63 \, \text{m}$
4. Time:
   • The time taken to travel from $P$ to $Q$ is 6 seconds.
   • The time taken to travel from $Q$ to $R$ is 2 seconds.

We are asked to calculate the acceleration of the particle.

Approach:

We'll use the equations of motion under constant acceleration. The first step is to apply the kinematic equation: $s = ut + \frac{1}{2} a t^2$ where:

• $s$ is the displacement,
• $u$ is the initial velocity,
• $t$ is the time, and
• $a$ is the acceleration.

Step 1: Motion from $P$ to $Q$

• Distance: $PQ = 105 \, \text{m}$,
• Time: $t_{PQ} = 6 \, \text{seconds}$.

Since the particle moves with uniform acceleration from rest:

• The velocity at $P$ is unknown, but we can use the equation: $PQ = u_P \cdot t_{PQ} + \frac{1}{2} a t_{PQ}^2$ Since $u_P$ is the velocity at point $P$, we need to calculate this using another relationship from $Q$.

Let me calculate the acceleration for you using these relationships.The acceleration of the particle is approximately $5.83 \, \text{m/s}^2$.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do we calculate the velocity at point $Q$?
2. How would we calculate the velocity of the particle at point $R$?
3. What is the total distance traveled by the particle from $O$ to $R$?
4. How does uniform acceleration affect the time intervals between points?
5. Could we calculate the time it takes to reach point $P$ from rest?

Tip: When dealing with uniform acceleration, kinematic equations are essential tools for connecting displacement, velocity, time, and acceleration.