Understanding and Calculating Uniform Acceleration in Physical Motion

Motion with uniform acceleration is a fundamental concept in physics. This article will guide you through the steps to calculate the acceleration of a body in motion using various methods, including the equations of motion and manipulation of kinematic equations.

Introduction to Uniform Acceleration

Uniform acceleration refers to a situation where an object's velocity changes at a constant rate. This means that the acceleration (the rate of change of velocity) remains constant throughout the motion. The kinematic equations are powerful tools for analyzing motion with uniform acceleration. In this article, we will demonstrate the process of finding the acceleration of a body given its distance covered in successive seconds.

Critical Equation for Uniform Acceleration

The key equation for uniform acceleration is the distance traveled in the nth second, which is given by:

s_n u frac{1}{2}a(2n - 1)

Here, s_n represents the distance covered in the nth second, u is the initial velocity, a is the acceleration, and n is the second.

Case Study: Calculating Acceleration

Consider a body moving with uniform acceleration that covers 6 meters in the first second and 10 meters in the second second. We need to determine the acceleration of the body. Let's break down the problem into manageable steps.

Step 1: Apply the Distance Formula for the First Second

For the first second (n 1):

s_1 u frac{1}{2}a(2 cdot 1 - 1) u frac{1}{2}a

Given s_1 6 meters, we can write:

u frac{1}{2}a 6 —— (1)

Step 2: Apply the Distance Formula for the Second Second

For the second second (n 2):

s_2 u frac{1}{2}a(2 cdot 2 - 1) u frac{3}{2}a

Given s_2 10 meters, we can write:

u frac{3}{2}a 10 —— (2)

Step 3: Subtract Equation (1) from Equation (2)

(u frac{3}{2}a) - (u frac{1}{2}a) 10 - 6

This simplifies to:

frac{3}{2}a - frac{1}{2}a 4

frac{2}{2}a 4

a 4text{ m/s}^2

Alternative Method Using Kinematic Equations

Another approach to solving this problem involves using the kinematic equation for uniformly accelerated motion, which is:

d ut frac{1}{2}at^2

Here, d is the displacement, u is the initial velocity, a is the acceleration, and t is the time.

Application to the Problem

For two seconds, the displacement d 9text{ m} and the time t 2text{ s}. Since the object starts from rest, the initial velocity u 0text{ m/s}. Plugging in the known values:

d ut frac{1}{2}at^2 Rightarrow 9 frac{1}{2}a(2)^2

9 2a Rightarrow a 4.5text{ m/s}^2

Summary

The problem involves calculating the uniform acceleration of a body given that it covers 6 meters in the first second and 10 meters in the second second. Using the equations of motion, we derived the acceleration to be 4 m/s2. An alternative method using the kinematic equation resulted in a slightly different but accurate answer of 4.5 m/s2. Both methods align with the fundamental principles of uniformly accelerated motion.

Conclusion

Motion with uniform acceleration is a significant concept in physics with wide-ranging applications in various fields. Understanding these principles and techniques is crucial for solving real-world problems involving motion under constant acceleration.