A particle starts from rest with a constant acceleration. At a time t second, the speed is found to be 100 m/s and one second later the speed becomes 150 m/s. The value of acceleration and the distance travelled during the (t + 1)th second will be respectively :

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the given information and apply the principles of kinematics.

### Given Information:
1. The particle starts from rest, so initial velocity $ u = 0 $.
2. At time $ t $, the speed is 100 m/s.
3. At time $ t + 1 $, the speed is 150 m/s.

### Step-by-Step Explanation:

#### 1. Determine the Acceleration:
The change in speed from time $ t $ to $ t + 1 $ is from 100 m/s to 150 m/s. This change occurs over a period of 1 second.

- Change in speed = Final speed - Initial speed = 150 m/s - 100 m/s = 50 m/s
- Time interval = 1 second

Using the formula for acceleration $ a $:
$$ a = \frac{\text{Change in speed}}{\text{Time interval}} = \frac{50 \, \text{m/s}}{1 \, \text{s}} = 50 \, \text{m/s}^2 $$

Thus, the acceleration is $ 50 \, \text{m/s}^2 $.

#### 2. Calculate the Distance Travelled During the $(t + 1)$th Second:
To find the distance travelled during the $(t + 1)$th second, we use the formula for distance travelled in the nth second:
$$ \text{Distance in } n\text{th second} = u + \frac{a}{2}(2n - 1) $$

For the $(t + 1)$th second:
- Initial velocity at time $ t $ is 100 m/s (since speed at time $ t $ is 100 m/s).
- Acceleration $ a = 50 \, \text{m/s}^2 $.

Plugging in the values:
$$ \text{Distance in } (t + 1)\text{th second} = 100 + \frac{50}{2}(2 \times 1 - 1) $$
$$ = 100 + 25 \times 1 $$
$$ = 100 + 25 $$
$$ = 125 \, \text{m} $$

### Conclusion:
- The acceleration is $ 50 \, \text{m/s}^2 $.
- The distance travelled during the $(t + 1)$th second is $ 125 \, \text{m} $.

Therefore, Option 1: $ 50 \, \text{m/s}^2 \text{ and } 125 \, \text{m} $ is the correct answer.

### Explanation of Incorrect Options:

- **Option 2: $ 25 \, \text{m/s}^2 \text{ and } 125 \, \text{m} $**:
  - The acceleration calculated is $ 50 \, \text{m/s}^2 $, not $ 25 \, \text{m/s}^2 $.

- **Option 3: $ 50 \, \text{m/s}^2 \text{ and } 200 \, \text{m} $**:
  - While the acceleration is correct, the distance travelled during the $(t + 1)$th second is $ 125 \, \text{m} $, not $ 200 \, \text{m} $.

- **Option 4: $ 75 \, \text{m/s}^2 \text{ and } 200 \, \text{m} $**:
  - Both the acceleration and the distance are incorrect based on the calculations.

Thus, Option 1 is the only correct choice based on the given information and calculations.