derivation for distance travelled in nth second by a uniformly accelerated body by graphical method

To derive the formula for the distance traveled by a uniformly accelerated body during the nth second using a graphical method, we can visualize the motion on a distance-time graph and a velocity-time graph. Let's break down the steps in a clear and logical manner.

Understanding Uniform Acceleration

A body moving with uniform acceleration has a constant rate of change of velocity over time. This means that if you were to plot its velocity against time, you would get a straight line, indicating that the velocity increases linearly.

Distance-Time and Velocity-Time Graphs

Start by considering the velocity-time graph for a uniformly accelerated object. The area under the curve of this graph represents the distance traveled. Here’s how we can derive the distance traveled in the nth second:

• Let u be the initial velocity of the body.
• a is the constant acceleration.
• At the end of n seconds, the final velocity v can be expressed as v = u + an.

Calculating Distance Traveled in n Seconds

The total distance traveled in the first n seconds can be given by the formula:

S_n = un + (1/2)an².

This equation represents the distance covered in the first n seconds. However, to find the distance traveled specifically in the nth second, we need to calculate the difference between the total distance covered in n seconds and that covered in (n-1) seconds.

Finding Distance in the nth Second

We can express the distance traveled in the nth second as:

S_nth = S_n - S_(n-1).

Now, substituting the formula for total distance:

• S_n = un + (1/2)an²
• S_(n-1) = u(n-1) + (1/2)a(n-1)²

Now, expanding the equation for S_(n-1):

S_(n-1) = un - u + (1/2)a(n² - 2n + 1).

Putting it together:

S_nth = (un + (1/2)an²) - [un - u + (1/2)a(n² - 2n + 1)].

Simplifying the Expression

Now let's simplify:

• Cancel out un:
• S_nth = (1/2)an² + u - (1/2)a(n² - 2n + 1)
• This reduces to:
• S_nth = u + (1/2)a(2n - 1).

Final Formula for Distance in nth Second

Hence, the distance traveled by a uniformly accelerated body during the nth second is:

S_nth = u + (1/2)a(2n - 1).

Conclusion and Application

This formula allows you to calculate the distance an object covers during any specific second of its uniformly accelerated motion. It has practical applications in various fields, such as physics and engineering, where understanding motion is crucial.