The angular position of a point on the rim of a 45.8 cm rotating wheel is given by θ(t) = 3.6 t2 − 2.8 t +7.5, where θ is measured in radians and t is measured in seconds.
What is the instantaneous speed v of the point at time t = 6.5 s?
To find the instantaneous speed v of the point at time t = 6.5 s, we need to find the derivative of the angular position function θ(t) and evaluate it at t = 6.5 s.
Step 1: Find the derivative of θ(t)
Differentiate the function θ(t) with respect to t to find its derivative.
θ'(t) = d/dt (3.6 t^2 − 2.8 t + 7.5)
Differentiating each term using the power rule:
θ'(t) = (2)(3.6)(t)^(2-1) + (1)(-2.8)(t)^(1-1) + 0
= 7.2t - 2.8
Step 2: Evaluate the derivative at t = 6.5 s
Plug in t = 6.5 s into the derivative function θ'(t) to find the instantaneous speed.
v = θ'(6.5)
= 7.2(6.5) - 2.8
= 46.8 - 2.8
= 44 m/s
Therefore, the instantaneous speed v of the point at time t = 6.5 s is 44 m/s.