Calculating Distance Traveled by an Object with Uniform Acceleration

In this article, we will explore the mathematical derivation and practical application of calculating the distance traveled by an object moving with uniform acceleration during a specific interval, namely the 4th and 5th seconds. This is a fundamental concept in mechanics and helps us understand the behavior of objects under varying acceleration conditions.

Key Equations for Uniform Acceleration

The basic equations of motion for an object moving with uniform acceleration are:

s ut frac{1}{2}at^2 s_n un frac{1}{2}an^2

Where:

s represents the distance traveled in time t u is the initial velocity a is the uniform acceleration

Distance Traveled in the 4th and 5th Seconds

To determine the distance traveled in the interval between the 4th and 5th seconds, we will calculate the distances at t5 seconds and t4 seconds, and then find the difference between these distances.

Distance at 5 Seconds

The distance traveled in the first n seconds is given by:

s n un 1 2 a n 2

For t5 seconds:

s 5 5 u 1 2 a 5 2 25u 25a 2

Which simplifies to:

s 5 25 u 25 a 2

Distance at 4 Seconds

For t4 seconds:

s 4 4 u 1 2 a 4 2 16 u 16 a 2

Which simplifies to:

s 4 16 u 16 a 2

Distance Traveled Between 4th and 5th Seconds

The distance traveled between the 4th and 5th seconds is the difference between the distances at 5 and 4 seconds:

s 4 to 5 25 u 25 a - 16 u 16 a 2 9 u 9 a 2 9 u 9 a 2

This simplifies to:

s_{4 to 5} frac{9}{2}u frac{9}{2}a

Alternative Method Calculating Distance

Alternatively, you can use the method based on the average speed in the interval. The average speed in the 4th to 5th seconds is:

4.5 a u u a 2 2

The distance traveled in this interval is:

4.5 a · 1 4.5 a

Understanding Acceleration

Acceleration is defined as the rate of change of velocity with respect to time:

a dv/dt

In the case of free fall with no air resistance, the acceleration due to gravity is a constant value of 9.81 m/s2. Using Newton's equations of motion, we can calculate the distances and velocities at any given time.

Conclusion

Through the application of the equations of motion and the concept of uniform acceleration, we have derived the distance traveled by an object during the 4th and 5th seconds of its motion. This understanding is crucial for applications ranging from basic physics education to advanced engineering calculations.