How to Calculate the Distance Covered in the nth Second of Uniformly Accelerated Motion

Understanding the principles of uniformly accelerated motion is crucial for solving a variety of physics problems. One important aspect of this motion is determining the distance covered in the nth second. This article will delve into the formula for calculating such distances and provide a detailed derivation.

Formula for Distance Covered in the nth Second

The distance covered in the nth second of uniformly accelerated motion can be calculated using the following formula:

sn u frac12; a (2n - 1)

Where:

sn is the distance covered in the nth second u is the initial velocity (in meters per second) a is the constant acceleration (in meters per second squared) n is the time interval in seconds for which you want to find the distance

Derivation of the Formula

The formula for the distance covered in the nth second can be derived using the equations of motion. First, let's consider the total distance covered in n seconds:

Sn ut frac12; at2

For t n, this becomes:

Sn un frac12; an2

Next, let's find the total distance covered in (n-1) seconds:

Sn-1 (u(n-1)) frac12; a(n-1)2

The distance covered during the nth second is the difference between these two distances:

sn Sn - Sn-1

Substituting the expressions for Sn and Sn-1:

sn (un frac12; an2) - ((u(n-1)) frac12; a(n-1)2)

After simplifying the expression, you will arrive at:

sn u frac12; a (2n - 1)

Example Calculation

Consider an object starting with an initial velocity of 5 m/s and having an acceleration of 2 m/s2. To find the distance covered in the 3rd second:

s3 5 times; frac12; times; 2 times; 2 times; 3 - 1 5 times; frac12; times; 2 times; 5 5 times; 5 10 meters

This formula is essential for solving problems related to uniformly accelerated motion in physics, providing a clear and precise method for determining distances in specific time intervals.

Short Summary

The distance traveled in the nth second is given by the simplified formula:

Sn u frac12; a (2n - 1)

This is derived from the equations of motion and is a fundamental tool in analyzing uniformly accelerated motion.