A motor cyclist accelerate from rest with acceleration of `2 m//s^(2)` for a time of `10 sec`. Then he moves with a constant velocity for `20 sec` and then finally comes to rest with a deceleration of `1 m//s^(2)`. Average speed for complete journey is `:-`

Options

A. `10 m//s`
B. `14 m//s`
C. `20m//s`
D. `28 m//s`

Explanation

To solve the problem step by step, we will break down the journey of the motorcyclist into three segments: acceleration, constant velocity, and deceleration.

### Step 1: Calculate the distance during acceleration
The motorcyclist accelerates from rest with an acceleration of \(2 \, \text{m/s}^2\) for \(10 \, \text{s}\).

Using the formula for distance during acceleration:
\[ s_1 = ut + \frac{1}{2} a t^2 \]
where:
- \(u = 0 \, \text{m/s}\) (initial velocity),
- \(a = 2 \, \text{m/s}^2\) (acceleration),
- \(t = 10 \, \text{s}\) (time).

Substituting the values:
\[ s_1 = 0 \cdot 10 + \frac{1}{2} \cdot 2 \cdot (10)^2 = 0 + \frac{1}{2} \cdot 2 \cdot 100 = 100 \, \text{m} \]

### Step 2: Calculate the distance during constant velocity
After accelerating, the motorcyclist moves with a constant velocity for \(20 \, \text{s}\). The velocity at the end of the acceleration phase can be calculated as:
\[ v = u + at = 0 + 2 \cdot 10 = 20 \, \text{m/s} \]
Now, using this constant velocity to find the distance:
\[ s_2 = v \cdot t = 20 \cdot 20 = 400 \, \text{m} \]

### Step 3: Calculate the distance during deceleration
Finally, the motorcyclist decelerates at \(1 \, \text{m/s}^2\) until coming to rest. The initial velocity for this phase is \(20 \, \text{m/s}\).

Using the formula for distance during deceleration:
\[ s_3 = \frac{v^2 - u^2}{2a} \]
where:
- \(v = 0 \, \text{m/s}\) (final velocity),
- \(u = 20 \, \text{m/s}\) (initial velocity),
- \(a = 1 \, \text{m/s}^2\) (deceleration).

Substituting the values:
\[ s_3 = \frac{0^2 - (20)^2}{2 \cdot (-1)} = \frac{-400}{-2} = 200 \, \text{m} \]

### Step 4: Calculate total distance and total time
Now, we can find the total distance covered:
\[ s_{\text{total}} = s_1 + s_2 + s_3 = 100 + 400 + 200 = 700 \, \text{m} \]

The total time taken for the journey is:
\[ t_{\text{total}} = 10 + 20 + 20 = 50 \, \text{s} \]

### Step 5: Calculate average speed
Finally, the average speed can be calculated using the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{700 \, \text{m}}{50 \, \text{s}} = 14 \, \text{m/s} \]

Thus, the average speed for the complete journey is \(14 \, \text{m/s}\).

---