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.
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.
To find the velocity, we need to calculate the derivative of the position function with respect to time (t). So, let's differentiate the position function s(t) with respect to t.
The given position function is: s(t) = 2t^2 - 4t + 1(t - 1)^2
Let's expand and simplify the function first:
s(t) = 2t^2 - 4t + (t^2 - 2t + 1)
= 2t^2 - 4t + t^2 - 2t + 1
= 3t^2 - 6t + 1
Now, let's find the derivative of s(t) with respect to t:
v(t) = d/dt (s(t))
= d/dt (3t^2 - 6t + 1)
Differentiating each term separately, we get:
v(t) = 6t - 6
Therefore, the velocity function at any given time t is v(t) = 6t - 6 meters per second.
To find the velocity, v(t), at any given time, t, we need to find the derivative of the position function, s(t).
Given that s(t) = 2t^2 - 4t + 1(t - 1)^2, we can simplify it before taking the derivative.
First, let's expand (t - 1)^2:
(t - 1)^2 = (t - 1)(t - 1) = t^2 - 2t + 1
Now, replace (t - 1)^2 in the original equation:
s(t) = 2t^2 - 4t + 1(t - 1)^2
= 2t^2 - 4t + 1(t^2 - 2t + 1)
= 2t^2 - 4t + t^2 - 2t + 1
= 3t^2 - 6t + 1
Now, we can take the derivative of s(t) with respect to t:
v(t) = d(s(t))/dt = d(3t^2 - 6t + 1)/dt
To differentiate, we use the power rule: d(x^n)/dx = n*x^(n-1).
Applying the power rule, we get:
v(t) = 2 * 3t^(2-1) - 6 * 1t^(1-1) + 0
= 6t - 6
So, the velocity function, v(t), is given by v(t) = 6t - 6 meters per second.