Understanding Uniform Acceleration in Linear Motion: A Comprehensive Guide

When analyzing the motion of a body in a straight line with uniform acceleration, it is important to understand the parameters involved, such as distance traveled, initial velocity, and acceleration. This article will explore how to determine the acceleration of a body that travels 55 meters in the 8th second and 85 meters in the 13th second using the principles of uniform acceleration and kinematic equations.

Understanding Uniform Acceleration

Uniform acceleration refers to a situation where the acceleration of a body remains constant over time. In such cases, the kinematic equations can be used to solve for various parameters. These equations are derived from the basic principles of motion and are essential for analyzing linear motion problems.

Methods for Solving Uniform Acceleration Problems

There are several methods to solve uniform acceleration problems, depending on the given information. The primary methods include using the distance traveled in the nth second and the total distance traveled in a certain time. Both of these methods will be explained below using the given problem.

Solving Uniform Acceleration Using Distances in Specific Seconds

An efficient way to find the acceleration of a body moving in a straight line with uniform acceleration is by using the formula for the distance traveled during the nth second:

sn u a(2n - 1)/2

This formula is derived from the SUVAT equations, where:

sn is the distance traveled in the nth second, u is the initial velocity, a is the acceleration, n is the second in which the distance is calculated.

Given data for the problem:

s8 55 m (distance traveled in the 8th second) s13 85 m (distance traveled in the 13th second)

We can set up the equations for the 8th and 13th seconds:

Equation for the 8th Second:

55 u a(2(8) - 1)/2

Simplifying:

55 u a(15)/2

2(55) 2u 15a

110 2u 15a

(Equation 1)

Equation for the 13th Second:

85 u a(2(13) - 1)/2

Simplifying:

85 u a(25)/2

2(85) 2u 25a

170 2u 25a

(Equation 2)

Step-by-Step Solution

To solve these equations simultaneously:

Step 1: Subtract Equation 1 from Equation 2

170 - 110 (2u 25a) - (2u 15a)

60 10a

a 6 m/s2

Step 2: Substitute a back into Equation 1 to find u

110 2u 15(6)

110 2u 90

20 2u

u 10 m/s

Conclusion

The acceleration of the body is 6 m/s2 and the initial velocity is 10 m/s.

Alternative Method: Using Total Distance Traveled

Another method to solve this problem is by using the total distance traveled during a specific time interval. The kinematic equation for the total distance traveled in time t is:

s ut (1/2)at2

This equation can be used to find the total distance traveled in the 8th second and the 13th second, and the difference will give the required distance.

For the 8th second:

55 u (1/2)a(82 - 72)

55 u (1/2)a(64 - 49)

55 u (1/2)a(15)

For the 13th second:

85 u (1/2)a(132 - 122)

85 u (1/2)a(169 - 144)

85 u (1/2)a(25)

Subtracting these two equations:

85 - 55 (u (1/2)a(25)) - (u (1/2)a(15))

30 (1/2)a(25 - 15)

30 (1/2)a(10)

30 5a

a 6 m/s2

Substituting a back into the 8th second equation:

55 u (1/2)(6)(15)

55 u 45

u 10 m/s

Conclusion with Summary

Using both methods, we have determined that the acceleration of the body is 6 m/s2 and the initial velocity is 10 m/s. This problem demonstrates the application of kinematic equations in solving uniform acceleration problems and highlights the importance of understanding these principles.

By utilizing the right equations and step-by-step problem-solving techniques, you can efficiently analyze and solve uniform motion problems. Whether through distances in specific seconds or total distances, these methods provide a solid foundation for understanding linear motion with constant acceleration.

Keywords: uniform acceleration, linear motion, kinematic equations