Analyzing Distance Covered by a Particle Under Constant Acceleration

In this article, we will explore the distances covered by a particle that starts from rest and moves with a constant acceleration. Specifically, we will determine the relationship between the distances covered in the first 10 seconds and the last 10 seconds. We will use the classic kinematic equations to derive these relationships and understand the underlying principles.

Problem Statement

Given a particle starting from rest and moving with a constant acceleration, let S1 be the distance covered in the first 10 seconds and S2 be the distance covered in the last 10 seconds. The objective is to determine the condition under which S2 is a multiple of S1.

Derivation and Solution

Let's begin by applying the kinematic equation for distance traveled under constant acceleration:

S ut (1/2)at2

where:

u is the initial velocity (0 in this case since the particle starts from rest) a is the constant acceleration t is the time interval

Distance Covered in the First 10 Seconds

In the first 10 seconds, the initial velocity is 0 and the time is 10 seconds:

S1 0 × 10 (1/2)a × (102)

S1 50a

Distance Covered in the Second 10 Seconds

For the second 10 seconds, we need to determine the initial velocity. The final velocity (initial velocity for the next 10 seconds) at the end of the first 10 seconds is:

v u at

v 0 a × 10

v 10a

Thus, the initial velocity for the next 10 seconds is 10a. Using this initial velocity and the same constant acceleration, the distance covered in the second 10 seconds is:

S2 10a × 10 (1/2)a × (102)

S2 100a 50a

S2 150a

Comparing S2 and S1

To find the relationship between S2 and S1, we compare their values:

S2 150a

S1 50a

S2 / S1 150a / 50a

S2 / S1 3

Therefore, the condition is S2 3S1.

Conclusion

The problem demonstrates that under constant acceleration, the distance a particle covers in the second 10 seconds is three times the distance it covers in the first 10 seconds. This relationship can be expressed as S2 3S1.

Understanding this concept is crucial for solving more complex problems related to kinematics and is applicable in various fields such as physics, engineering, and sports science.