Finding Initial Velocity and Uniform Acceleration of a Body Moving with Uniform Acceleration

In this article, we will explore the concept of a body moving with uniform acceleration and how to calculate its initial velocity and acceleration using the equations of motion. Specifically, we will solve for the initial velocity and acceleration of a body that covers 20 meters in the 4th second and 30 meters in the 8th second.

Understanding the Problem

The problem involves finding the initial velocity u and the acceleration a of a body moving with uniform acceleration. We will use the equations of motion to solve this problem step by step.

The Equations of Motion

The distance covered by a body in the n-th second of its motion can be calculated using the formula:

S_n u dfrac{1}{2} a (2n - 1)

Where:

S_n is the distance covered in the n-th second. u is the initial velocity. a is the acceleration. n is the second number.

Setting Up the Equations

Given that the body covers 20 meters in the 4th second and 30 meters in the 8th second, we can set up the following equations:

S_4 u dfrac{1}{2} a (2 cdot 4 - 1) 20

S_8 u dfrac{1}{2} a (2 cdot 8 - 1) 30

These simplify to:

u dfrac{7}{2} a 20

u dfrac{15}{2} a 30

Solving the System of Equations

Step 1: Solve the System of Equations

To eliminate u, we can subtract the first equation from the second:

(u dfrac{15}{2} a) - (u dfrac{7}{2} a) 30 - 20

This simplifies to:

dfrac{15}{2} a - dfrac{7}{2} a 10

dfrac{8}{2} a 10

4a 10

a dfrac{10}{4} 2.5 , text{m/s}^2

Step 2: Substitute a back to find u

Now, substitute a 2.5 , text{m/s}^2 back into either equation. Using the first equation:

u dfrac{7}{2} cdot 2.5 20

u dfrac{7 cdot 2.5}{2} 20

u dfrac{17.5}{2} 20

u 8.75 20

u 20 - 8.75 11.25 , text{m/s}

Conclusion

Therefore, the initial velocity and the acceleration of the body are:

Initial Velocity u: 11.25 , text{m/s}

Acceleration a: 2.5 , text{m/s}^2

This solution demonstrates how to use the equations of motion to find the initial velocity and acceleration of a body moving with uniform acceleration.

Keywords: initial velocity, uniform acceleration, equations of motion