If an object is moving with constant acceleration along one direction only, then the distance travelled in the nth second is :
[Take U = Initial Velocity a = Acceleration]

Question

Accepted Answer

To determine the distance traveled by an object in the nth second when it is moving with constant acceleration, we use the formula for the distance traveled in the nth second, which is derived from the equations of motion. The formula is:

$$ s_n = u + \frac{a}{2}(2n - 1) $$

Where:
- $ s_n $ is the distance traveled in the nth second,
- $ u $ is the initial velocity,
- $ a $ is the constant acceleration,
- $ n $ is the specific second for which we are calculating the distance.

### Explanation for Option 2:

Option 2 states the formula as $ U + \frac{a}{2}(2n - 1) $. This matches the derived formula for the distance traveled in the nth second. The derivation is based on the equations of motion, specifically:

1. The total distance traveled in n seconds is given by:
   $$ S_n = un + \frac{1}{2}a n^2 $$

2. The total distance traveled in (n-1) seconds is:
   $$ S_{n-1} = u(n-1) + \frac{1}{2}a (n-1)^2 $$

3. The distance traveled in the nth second is the difference between these two:
   $$ s_n = S_n - S_{n-1} $$
   $$ s_n = \left(un + \frac{1}{2}a n^2\right) - \left(u(n-1) + \frac{1}{2}a (n-1)^2\right) $$

4. Simplifying this expression, we get:
   $$ s_n = u + \frac{a}{2}(2n - 1) $$

This confirms that Option 2 is indeed the correct formula for the distance traveled in the nth second.

### Explanation for why other options are incorrect:

- **Option 1: $ U + a (1 + n^2) $**
  - This formula does not correctly represent the distance traveled in the nth second. It incorrectly includes a term $ n^2 $ without the proper context of the difference between consecutive time intervals.

- **Option 3: $ U + \frac{a}{2}(2n + 1) $**
  - This formula is close but incorrect because it uses $ (2n + 1) $ instead of $ (2n - 1) $. The correct derivation shows that the term should be $ (2n - 1) $ to account for the difference in distance between the nth and (n-1)th seconds.

- **Option 4: $ U + \frac{a}{2}(2n + 1)^2 $**
  - This formula is incorrect because it squares the term $ (2n + 1) $, which is not part of the standard derivation for the distance traveled in the nth second. The squaring introduces an unnecessary complexity that does not align with the physics of constant acceleration.

In conclusion, Option 2 is the correct choice as it accurately represents the distance traveled in the nth second for an object moving with constant acceleration.