Displacement in the nth Second Formula | Complete Notes with Derivation

Displacement during the nth second under constant acceleration explained with formula and illustration.

-  Dr.Sanjaykumar Pawar

 nth Second Displacement Formula | Class 11 Physics Notes

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FAQ 

Q1. What is displacement in the nth second?
A. It is the displacement covered by an object during only the nth second of its motion.

Q2. What is the formula for displacement in the nth second?
A. The formula is:
Sₙₜₕ = u + (a/2)(2n−1)

Q3. When can this formula be used?
A. It is applicable only when the acceleration is constant.

Q4. Is nth second displacement the same as total displacement?
A. No. Total displacement is measured from the start of motion, whereas nth second displacement is measured only during one specific second.

Q5. Which exams include questions on the nth second formula?
A. This topic is important for Class 11 Physics, JEE Main, JEE Advanced, NEET, and other competitive examinations. 

Displacement in the n^th Second

1. What is Displacement in the n^th Second?

Displacement in the n^th second means the displacement covered during only one particular second.

Examples

• 1st second → 0 s to 1 s
• 2nd second → 1 s to 2 s
• 3rd second → 2 s to 3 s
• 5th second → 4 s to 5 s

It does not mean the total displacement from the starting point.

2. Total Displacement After n Seconds

The displacement after n seconds is given by:

S_n = un + ½ an²

Where:

• u = Initial velocity
• a = Constant acceleration
• n = Time in seconds
• S_n = Total displacement after n seconds

3. Total Displacement After (n−1) Seconds

The displacement up to the end of the previous second is:

S_n−1 = u(n−1) + ½ a(n−1)²

This represents the total displacement before the n^th second begins.

4. Formula for Displacement in the n^th Second

Displacement during only the n^th second is:

S_nth = S_n − S_n−1

This means:

Displacement in one interval = Total displacement at the end − Total displacement at the beginning.

5. Substitute the Values

S_nth = (un + ½an²) − [u(n−1) + ½a(n−1)²]

Now simplify the expression.

6. Final Formula

S_nth = u + (a/2)(2n − 1)

This is the standard formula used to calculate displacement during the n^th second.

Meaning of Symbols

Symbol	Meaning
Snth	Displacement during the nth second
u	Initial velocity
a	Constant acceleration
n	Number of the second

Example

Given:

• u = 5 m/s
• a = 2 m/s²
• Find displacement in the 4th second.

S_4th = 5 + (2/2)(2×4 −1)

= 5 + 7

= 12 m

Answer: The object moves 12 m during the 4th second.

Important Points

• Acceleration must remain constant.
• The formula gives displacement, not total displacement.
• The time interval is always 1 second.
• The n^th second starts at (n−1) s and ends at n s.
• Examples:
  • 3rd second → 2 s to 3 s
  • 6th second → 5 s to 6 s
  • 10th second → 9 s to 10 s

Total Displacement vs Displacement in the n^th Second

Total Displacement	Displacement in nth Second
Measured from 0 s to n s	Measured only during one second
S = ut + ½at²	Snth = u + (a/2)(2n−1)
Gives total position change	Gives position change in one specific second

Quick Revision

1. Find total displacement after n seconds.
2. Find total displacement after (n−1) seconds.
3. Subtract both values.
4. Use the final formula:

S_nth = u + (a/2)(2n−1)

Memory Trick:

n^th second = Total displacement till n − Total displacement till (n−1).

NEET Practice Questions – Displacement in the n^th Second

Formula: S_nth = u + ^a⁄₂(2n − 1)

Type 1: Direct MCQs

Q1 (Easy)
Displacement in the n^th second is:

A. u + a/2(2n+1)      B. u + a/2(2n−1)      C. ut + ½at²      D. un + ½an²

Answer: B

Q2
u = 8 m/s, a = 4 m/s². Find displacement in 3rd second.

Answer: 18 m

Q3
Starts from rest, a = 6 m/s². Displacement in 5th second?

Answer: 27 m

Q4
6th second displacement = 29 m, a = 2 m/s². Find u.

Answer: 18 m/s

Q5 (Hard)
8th second displacement = 40 m, a = 4 m/s². Find u.

Answer: 10 m/s

Type 2: Statement Based

Q6
Statement I: S_nth = S_n − S_(n−1).
Statement II: nth-second displacement equals difference of total displacements.

Answer: Both True.

Q7
Statement I: Formula works for variable acceleration.
Statement II: Constant acceleration is required.

Answer: I False, II True.

Q8
Statement I: Formula gives displacement.
Statement II: Distance and displacement are always equal.

Answer: I True, II False.

Type 3: Assertion & Reason

Q9
Assertion (A): 4th-second displacement = S₄ − S₃.
Reason (R): 4th second is from 3 s to 4 s.

Answer: Both true and R explains A.

Q10
Assertion (A): Formula works for changing acceleration.
Reason (R): Derivation assumes constant acceleration.

Answer: A False, R True.

Q11
Assertion (A): If u = 0, S_nth = (a/2)(2n−1).
Reason (R): Body starts from rest.

Answer: Both true and R explains A.

Type 4: Match the Columns

Column I	Column II
A. 1st second	P. 4–5 s
B. 2nd second	Q. 1–2 s
C. 5th second	R. 0–1 s
D. 8th second	S. 7–8 s

Correct Match: A-R, B-Q, C-P, D-S

Type 5: Diagram Based

Q12

0s ---- 1 ---- 2 ---- 3 ---- 4
4th second = 3 s to 4 s

Answer: 3 s to 4 s

Q13

S
|
|      *
|   *
| *
+---------------> t

Correct equation: S = ut + ½at²

Answer Key

1-B, 2-18 m, 3-27 m, 4-18 m/s, 5-10 m/s, 6-Both True, 7-I False, II True, 8-I True, II False, 9-Both true and R explains A, 10-A False, R True, 11-Both true and R explains A, 12-3 s to 4 s, 13-S = ut + ½at²

Column Match: A-R, B-Q, C-P, D-S