A particle starting from rest has a constant acceleration of `4 m//s^2` for 4 s. It then retards uniformly for next 8 s and comes to rest. Find during the motion of particle (a) average acceleration (b) average speed and (c) average velocity.

Explanation

To solve the problem step by step, we will break it down into parts as follows:

### Given Data:
- Initial velocity (u) = 0 m/s (the particle starts from rest)
- Acceleration (a) = 4 m/s² for the first 4 seconds
- Time of acceleration (t₁) = 4 s
- Retardation (negative acceleration) = 2 m/s² for the next 8 seconds (derived from the problem)
- Time of retardation (t₂) = 8 s

### Step 1: Calculate the distance traveled during acceleration (s₁)
We can use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]

Substituting the known values:
\[ s₁ = 0 \cdot 4 + \frac{1}{2} \cdot 4 \cdot (4)^2 \]
\[ s₁ = 0 + \frac{1}{2} \cdot 4 \cdot 16 \]
\[ s₁ = 2 \cdot 16 \]
\[ s₁ = 32 \text{ meters} \]

### Step 2: Calculate the distance traveled during retardation (s₂)
Using the equation of motion:
\[ s = vt - \frac{1}{2} a t^2 \]

First, we need to find the final velocity (v) at the end of the acceleration phase:
\[ v = u + at \]
\[ v = 0 + 4 \cdot 4 \]
\[ v = 16 \text{ m/s} \]

Now, substituting into the equation for s₂:
\[ s₂ = 16 \cdot 8 - \frac{1}{2} \cdot 2 \cdot (8)^2 \]
\[ s₂ = 128 - \frac{1}{2} \cdot 2 \cdot 64 \]
\[ s₂ = 128 - 64 \]
\[ s₂ = 64 \text{ meters} \]

### Step 3: Calculate total distance traveled
Total distance (S) is the sum of s₁ and s₂:
\[ S = s₁ + s₂ \]
\[ S = 32 + 64 \]
\[ S = 96 \text{ meters} \]

### Step 4: Calculate average acceleration (a_avg)
Average acceleration is given by:
\[ a_{avg} = \frac{V_f - V_i}{T} \]
Where:
- \( V_f \) = final velocity = 0 m/s (comes to rest)
- \( V_i \) = initial velocity = 0 m/s
- \( T \) = total time = t₁ + t₂ = 4 + 8 = 12 s

Substituting the values:
\[ a_{avg} = \frac{0 - 0}{12} = 0 \text{ m/s}^2 \]

### Step 5: Calculate average speed (v_avg)
Average speed is given by:
\[ v_{avg} = \frac{\text{Total distance}}{\text{Total time}} \]
\[ v_{avg} = \frac{96}{12} \]
\[ v_{avg} = 8 \text{ m/s} \]

### Step 6: Calculate average velocity (v_avg)
Since the particle returns to rest at the starting point, the average velocity is:
\[ v_{avg} = \frac{\text{Total displacement}}{\text{Total time}} \]
Since the total displacement is also 96 meters (the same as total distance in this case):
\[ v_{avg} = \frac{96}{12} = 8 \text{ m/s} \]

### Final Answers:
(a) Average acceleration = 0 m/s²
(b) Average speed = 8 m/s
(c) Average velocity = 8 m/s