A particle starts from rest and accelerates uniformly along a straight line. The ratio of the distance it travels in the N^ th second to the distance it travels in the (N+1)^ th second is 5:7 . If the distance traveled in the N^ th second is x times the distance traveled in the first second, the value of x is ______.

Step-by-step solution

Let the uniform acceleration be a . The distance traveled in the n^ th second by a particle starting from rest is given by S_n = a 2 (2n - 1) . For the N^ th second, S_N = a 2 (2N - 1) . For the (N+1)^ th second, S_ N+1 = a 2 (2(N+1) - 1) = a 2 (2N + 1) . Given the ratio is 5:7 , we have: 2N - 1 2N + 1 = 5 7 14N - 7 = 10N + 5 4N = 12 N = 3 The distance traveled in the first second ( n=1 ) is S₁ = a 2 (2(1) - 1) = a 2 . The distance traveled in the N^ th second ( N=3 ) is S₃ = a 2 (2(3) - 1) = 5a 2 . The ratio x = S