Achieving the Ideal Ratio of Distance Covered in the nth Second to That Covered in n Seconds

Introduction to Acceleration and Distance Formulas:

In physics, when a car starts from rest and moves with constant acceleration, the distances covered at different instants provide critical information for various applications, including automotive testing and engineering design. This article delves into the mathematical derivation to find the ratio of the distance covered in the nth second to the distance covered in n seconds. We will primarily use the equations of motion with constant acceleration to derive the desired ratio.

Distance Covered in n Seconds (Sn)

The formula for the distance covered under constant acceleration a after n seconds starting from rest is given by:

S n frac{1}{2} a n^2

Distance Covered in the nth Second (Sn)

The distance covered in the nth second can be calculated using the formula:

S n S_n - S_{n-1}

where Sn-1 is the distance covered in n-1 seconds:

S n - 1 frac{1}{2} a (n-1)^2 frac{1}{2} a n^2 - 2a n a

Substituting Sn-1 into the equation:

S n frac{1}{2} a n^2 - (frac{1}{2} a n^2 - 2a n a) 2a n - a

Thus, the distance covered in the nth second simplifies to:

S n a n - frac{1}{2}

Ratio of distances

Now, we can find the ratio of the distance covered in the nth second to that covered in n seconds:

text{Ratio} frac{S_n}{S_n} frac{a n - frac{1}{2}}{frac{1}{2} a n^2}

Simplifying the ratio:

text{Ratio} frac{2n - 1}{n^2}

Thus, the ratio of the distance covered in the nth second to that covered in n seconds is:

frac{2n - 1}{n^2}

Verification Through Another Method

This ratio can also be derived through another method using the second equation of linear motion with constant acceleration:

S frac{1}{2} a n^2

For the distance covered in n-1 seconds:

S_{n-1} frac{1}{2} a (n-1)^2 frac{1}{2} a n^2 - an frac{1}{2}

The distance covered in the nth second is the difference between these two distances:

Delta S S - S_{n-1} frac{1}{2} a n^2 - (frac{1}{2} a n^2 - an frac{1}{2}) an - frac{1}{2}

Velocity at the End of n-1 seconds and Average Velocity

The velocity at the end of n-1 seconds is:

v_{n-1} a (n-1)

The average velocity in the nth second is:

v_{avg} frac{v_{n-1} v_n}{2} frac{a (n-1) a n}{2} frac{a (2n-1)}{2}

The distance covered in the nth second is:

S_n v_{avg} times 1 frac{a (2n-1)}{2}

The distance covered in n seconds is:

S frac{1}{2} a n^2

The ratio is thus:

text{Ratio} frac{2n-1}{n^2}

Conclusion

The derived ratio, (frac{2n - 1}{n^2}), offers a clear understanding of how distance covered in the nth second compares to the distance covered in n seconds for a car starting from rest and moving with constant acceleration. This ratio is fundamental for analyzing motion in kinematics and can be particularly useful in engineering, automotive testing, and physics education.

Keywords: Constant Acceleration, Distance Formula, nth Second Theorem