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# Distance Traveled Formula

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Last updated date: 11th Sep 2024
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## Total Distance Traveled Formula

Distance traveled describes how much path an object has covered in order to reach its destination in a specified time period. The distance covered formula for distance traveled is given as:

$d = vt$

Where,

$d$ = the distance traveled

$v$ = the velocity

$t$ = time taken to travel

Similar to the distance traveled formula, there is the distance traveled in the last second formula and formula for displacement which we will learn below.

## Uses and Applications of Distance Traveled Formula

The distance-covered formula is applicable to compute the distance of driving a car or swimming stretch in a pool. While driving a car, the distance will be computed in kilometers or miles, the rate is in kilometers per hour or miles per hour, and time is in hours. While swimming laps in a pool, the distance is computed in laps.

## Displacement in nth Second Formula

In order to calculate the displacement (position shift) from the velocity function, you just need to integrate the function. The negative areas below the x-axis subtracted from the total displacement. For this, we use a formula for displacement in nth second.

Displacement $= \int b_a v(t) dt$

In order to calculate the distance traveled we need to use the absolute value.

The displacement in the nth second distance formula covered in the nth second is given by $S_n = u + a(n - 12)$.

## Derive the Expression for the Formula for Displacement in the nth Second

• A is the acceleration

• V is the velocity

• U is the initial velocity

• S is the distance

$s = ut + \dfrac{1}{2}at^2$

In order to calculate the distance traveled at the time of the nth second, we compute the distance covered in n seconds and subtract (minus-) the distance covered in (n-1) seconds and obtain:-

$s = un - u(n-1) + \dfrac{1}{2}an^2 - \dfrac{1}{2}a(n-1)^2$

Simplifying provides us

$u \text{ term }= u(n-(n-1)) = u \times 1 = u$

$(n-1)^2 = n^2-2n+1$

$a \text{ term }= \dfrac{1}{2}a(n^2-(n-1)^2) = \dfrac{1}{2}a(n^2-n^2+2n-1)$ the $n^2$'s cancel and provide us $\dfrac[1}{2a}(2n-1)$.

The final distance traveled equation for displacement in the nth second is