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Question

# Write the formula for distance traveled in nth second.

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Solution

## Step 1: Given$S=ut+\frac{1}{2}a{t}^{2}$Here,$S$ is the distance traveled.$u$ is the initial velocity.$t$ is the time. $a$ is the accelerationStep 2: For distance traveled in n and n-1 secondLet,Initial velocity = $u$Acceleration = $a$Distance traveled in n second = ${S}_{n}$By the second equation of motion,Distance traveled in $t$ seconds,$S=ut+\frac{1}{2}a{t}^{2}$Distance traveled in n seconds,${S}_{n}=un+\frac{1}{2}a{n}^{2}$ ----- (i)Distance traveled in (n-1) seconds,${S}_{\left(n-1\right)}=u\left(n-1\right)+\frac{1}{2}a{\left(n-1\right)}^{2}$ ----- (ii)Step 3: Formula for distance traveled in n and n-1 secondTo calculate the distance traveled in nth second,${S}_{{n}^{th}}={S}_{n}-{S}_{\left(n-1\right)}$ ----- (iii)Putting values of ${S}_{n}$ and ${S}_{\left(n-1\right)}$ from equations (i) and (ii)${S}_{{n}^{th}}=un+\frac{1}{2}a{n}^{2}-\left[u\left(n-1\right)+\frac{1}{2}a{\left(n-1\right)}^{2}\right]$Simplifying the above equation,${S}_{{n}^{th}}=u+\frac{1}{2}a\left(2n-1\right)\phantom{\rule{0ex}{0ex}}{S}_{{n}^{th}}=u+a\left(n-\frac{1}{2}\right)$Therefore, the equation for distance traveled in nth second = ${S}_{{n}^{th}}=u+a\left(n-\frac{1}{2}\right)$.

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