A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and speed when t = 9.
f(t) = t ^-1 - t
plzzz help!!!
need help
To find the velocity and speed when t = 9, we need to first determine the derivative of the equation of motion, s = f(t), with respect to t. The derivative of f(t) will give us the velocity of the particle at any given time.
Let's start by finding the derivative of f(t):
f'(t) = d/dt (t^(-1) - t)
To find the derivative of t^(-1), we can use the power rule for derivatives:
d/dt (t^(-1)) = -1 * t^(-1-1) = -t^(-2) = -1/t^2
For the second term, d/dt (-t) = -1
Therefore, the derivative of f(t) is:
f'(t) = -1/t^2 - 1
Now, we can find the velocity when t = 9 by substituting t = 9 into f'(t):
v = f'(9) = -1/(9^2) - 1
To find the speed of the particle at t = 9, we need to take the absolute value of velocity because speed is always positive:
speed = |v| = |-1/(9^2) - 1|
Now we can calculate the values of velocity and speed.