Solving the Problem of a Particle with Uniform Acceleration

When dealing with the motion of a particle under uniform acceleration, it's essential to have a solid understanding of the equations of motion. This article will guide you through the process of solving a specific problem involving a particle moving in a straight line with uniform acceleration. The scenario involves distances covered in two consecutive 4-second intervals, and our goal is to determine the initial velocity of the particle.

Understanding the Problem

A particle is moving with a uniform acceleration and travels 24 meters in the first 4 seconds and 64 meters in the next 4 seconds. We need to find the initial velocity of the particle. This problem assumes the particle moves in a straight line and under uniform acceleration, but the discussion will cover the nuances of the scenario as well.

Equations of Motion

To solve this problem, we will use the equations of motion. The distance traveled ((s)) by an object under uniform acceleration can be expressed as:

(s ut frac{1}{2}at^2) (s) is the distance traveled, (u) is the initial velocity, (a) is the acceleration, (t) is the time.

Let's break down the problem step-by-step:

Step 1: Express the Known Distances Using the Equations of Motion

For the first 4 seconds:

(s_1 u cdot 4 frac{1}{2}a cdot 4^2)

Substituting the given distance (s_1 24) meters:

24 4u 8a

For the next 4 seconds (from 4 to 8 seconds), the total distance traveled in 8 seconds:

(s_{total} u cdot 8 frac{1}{2}a cdot 8^2)

The distance traveled in the second interval can be expressed as:

(s_2 s_{total} - s_1)

Substituting the total distance:

(s_2 (8u 32a) - (4u 8a))

(s_2 4u 24a)

Given (s_2 64) meters:

64 4u 24a

Step 2: Solve the System of Equations

We now have two equations:

(4u 8a 24) (4u 24a 64)

Subtract the first equation from the second:

((4u 24a) - (4u 8a) 64 - 24)

(16a 40)

(a frac{40}{16} 2.5 , text{m/s}^2)

Substitute (a 2.5 , text{m/s}^2) back into the first equation:

4u 8(2.5) 24

(4u 20 24)

(4u 4)

(u 1 , text{m/s})

Conclusion

The initial velocity of the particle is 1 m/s. This solution assumes the particle is moving in a straight line under uniform acceleration, which aligns with the given conditions.

While the problem description is clear, it's worth noting that the particle presumably traveled one distance for each consecutive interval, making the total of three distances. However, only two distances are given, which doesn't affect the solution methodology. The centripetal acceleration discussion pertains to a different context and doesn't impact the straight-line uniform acceleration scenario addressed here.

Related Keywords

acceleration equations of motion initial velocity

Further Reading and Resources

To deepen your understanding of uniform acceleration and equations of motion, consider exploring the following resources:

Physics Classroom - Equations of Motion Better Explained - Understanding Equations of Motion

These resources offer a more in-depth, visual, and intuitive approach to understanding the motion of particles under various conditions.