A particle moves along a line so that its position at any time, t>1 , is given by the function s(t)=2t2−4t+1(t−1)2 , where s is measured in meters and t is measured in seconds. Find the velocity, v(t) , at any given time, t.

Answer 1

To find velocity, we need to take the derivative of the position function, s(t), with respect to time, t.

s(t) = 2t^2 - 4t + 1 / (t - 1)^2

Taking the derivative:

s'(t) = d/dt (2t^2 - 4t + 1 / (t - 1)^2)
= (4t - 4) / (t - 1)^2 + (2t^2 - 4t + 1) * -2 / (t - 1)^3 [Using the quotient rule to differentiate (t - 1)^2]
= (4t - 4) / (t - 1)^2 - 2(2t^2 - 4t + 1) / (t - 1)^3

Simplifying:

s'(t) = (4t - 4 - 4t^2 + 8t - 2) / (t - 1)^3
= (-4t^2 + 12t - 5) / (t - 1)^3

Therefore, the velocity function is v(t) = (-4t^2 + 12t - 5) / (t - 1)^3.