Question

show that for a body moving with uniform acceleration ,the distance covered by it in the n^th second is given by S_n^th =ut +a/2(2n-1)

Answer 1

Its easy to derive

let us assume that the initial velocity of the object be 'u' and it moves under constant acceleration 'a'.

Now,

let S_n-1 be the distance travelled by the object in (n-1) seconds

let S_n be the distance travelled by the article in n seconds

so, the distance travelled in n^th second will be

D_n = S_n - S_n-1 (1)

we also know that

s = ut + (1/2)at²

so,

S_n = un + (1/2)an² (2)

S_n-1 = u(n-1) + (1/2)a(n-1)2 (3)

so, (1) becomes

D_n = un + (1/2)an² - u(n-1) + (1/2)a(n-1)²

or

D_n = un = (1/2)an² - un + u - (1/2)an² + an - (a/2)

thus, by solving further, we get

the distance travelled in nth second of uniformly accelerated motion

D _n = u + (a/2)[2n - 1]