Learn the concept of distance traveled in the nth second of motion with formulas, step-by-step derivations, and solved examples for JEE, NEET, and CBSE Class 11 Physics.
In kinematics, we often calculate the total displacement of a moving object over a given time. However, it is also useful to find the displacement in a particular second, such as the 5th or 10th second. The formula for displacement in the nth second helps us determine this.
Formula for Displacement in nth Second
The displacement of a particle moving with uniform acceleration in n seconds is given by: $$S_n = u n + \frac{1}{2} a n^2$$
The displacement in (n – 1) seconds is: $$S_{n-1} = u (n-1) + \frac{1}{2} a (n-1)^2$$
So, the displacement in the nth second is: $$S_n – S_{n-1} = u + \frac{a}{2} (2n – 1)$$
$$D_n = u + \frac{a}{2} (2n – 1)$$
This is the equation to find the displacement in the nth second.
Derivation for Displacement Travelled in nth Second of its Motion
1. Displacement in n seconds
The total displacement covered in n seconds under uniform acceleration is: $$S_n = u n + \frac{1}{2} a n^2$$
where:
- u = Initial velocity
- a = Acceleration
- n = Time in seconds
2. Displacement in (n-1) seconds
The displacement covered in (n – 1) seconds is: $$S_{n-1} = u (n – 1) + \frac{1}{2} a (n – 1)^2$$
3. Finding displacement in nth second
The displacement in the nth second is: $$D_n = S_n – S_{n-1}$$
Substituting values: $$D_n = \left( u n + \frac{1}{2} a n^2 \right) – \left( u (n – 1) + \frac{1}{2} a (n – 1)^2 \right)$$
Expanding, $$D_n = u n + \frac{1}{2} a n^2 – u (n – 1) – \frac{1}{2} a (n^2 – 2n + 1)$$
Simplifying, $$D_n = u + \frac{a}{2} (2n – 1)$$
This is the displacement in the nth second.
Question-Answer Format for JEE, NEET, and CBSE Class 11
Q1: What is meant by displacement in the nth second?
Ans: Displacement in the nth second refers to the distance covered by a body in only the nth second of motion, not the total displacement up to that time.
Q2: Why is the formula for displacement in the nth second useful?
Ans: It helps in solving kinematics problems efficiently, especially when finding how far an object moves in a particular second.
Q3: Can displacement in the nth second be negative?
Ans: Yes, if the object is moving in the negative direction (opposite to the initial direction of motion), then the displacement in the nth second can be negative.
MCQs with Answers and Explanations
MCQ 1
A body is thrown vertically upwards with an initial velocity of 40 m/s. What is the displacement in the 4th second? (Take $g = 10 \text{ m/s}^2$)
(a) 5 m
(b) 10 m
(c) 15 m
(d) 20 m
Answer: (b) 10 m
Explanation:
Using the formula: $D_n = u + \frac{a}{2} (2n – 1)$
For upward motion, acceleration $$a = -g = -10 \text{ m/s}^2$$ $$D_4 = 40 + \frac{-10}{2} (2 \times 4 – 1)$$
$$D_4 = 40 – 5 \times 7 = 40 – 35 = 10 \text{ m}$$
Do You Know?
- The equation for displacement in the nth second is derived from the second equation of motion.
- The displacement in the first second is always equal to the initial velocity u.
- If acceleration is zero, then displacement in the nth second remains constant.
- This concept is frequently used in projectile motion problems.
Worksheet
Solve the Following
- A car starts from rest and accelerates uniformly at 2 m/s². Find the displacement in the 6th second.
- A ball is thrown vertically upwards with a velocity of 50 m/s. Find the displacement in the 5th second.
- An object is moving with an initial velocity of 20 m/s and acceleration 3 m/s². Calculate its displacement in the 8th second.
Test Paper (10 Marks)
Section A: Conceptual Questions (4 Marks)
- Define displacement in the nth second. (2 Marks)
- How does acceleration affect the displacement in the nth second? (2 Marks)
Section B: Numerical Questions (6 Marks)
- A car starts from rest and accelerates uniformly at 5 m/s². Find the displacement in the 7th second. (3 Marks)
- A stone is dropped from a cliff. Find its displacement in the 3rd second. (Take $g = 10 \text{ m/s}^2$) (3 Marks)
Important Points for Quick Revision
✅ Formula: $D_n = u + \frac{a}{2} (2n – 1)$
✅ Displacement in the nth second is useful in kinematics and projectile motion.
✅ If acceleration is zero, displacement in each second is the same.
✅ When thrown upwards, acceleration is negative, reducing displacement over time.
✅ When dropped from a height, displacement follows increasing values due to acceleration due to gravity.
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