Answer
(a) (i) $v_{ave} = 6.0~cm/s$
(ii) $v_{ave} = -4.712~cm/s$
(iii) $v_{ave} = -6.134~cm/s$
(iv) $v_{ave} = -6.268~cm/s$
(b) We could estimate that the instantaneous velocity at $t = 1$ is $-6.3~cm/s$
Work Step by Step
(a) We can find the displacement at $t=1$:
$s = 2~sin~\pi t+ 3~cos~\pi t$
$s = 2~sin~\pi (1)+ 3~cos~\pi (1)$
$s = -3~cm$
(i) We can find the displacement at $t=2$:
$s = 2~sin~\pi t+ 3~cos~\pi t$
$s = 2~sin~\pi (2)+ 3~cos~\pi (2)$
$s = 3~cm$
We can find the average velocity:
$v_{ave} = \frac{3~cm-(-3~cm)}{2~s-1~s} = 6.0~cm/s$
(ii) We can find the displacement at $t=1.1$:
$s = 2~sin~\pi t+ 3~cos~\pi t$
$s = 2~sin~\pi (1.1)+ 3~cos~\pi (1.1)$
$s = -3.471203537635~cm$
We can find the average velocity:
$v_{ave} = \frac{-3.471203537635~cm-(-3~cm)}{1.1~s-1~s} = -4.712~cm/s$
(iii) We can find the displacement at $t=1.01$:
$s = 2~sin~\pi t+ 3~cos~\pi t$
$s = 2~sin~\pi (1.01)+ 3~cos~\pi (1.01)$
$s = -3.06134~cm$
We can find the average velocity:
$v_{ave} = \frac{-3.06134~cm-(-3~cm)}{1.01~s-1~s} = -6.134~cm/s$
(iv) We can find the displacement at $t=1.001$:
$s = 2~sin~\pi t+ 3~cos~\pi t$
$s = 2~sin~\pi (1.001)+ 3~cos~\pi (1.001)$
$s = -3.006268~cm$
We can find the average velocity:
$v_{ave} = \frac{-3.006268~cm-(-3~cm)}{1.001~s-1~s} = -6.268~cm/s$
(b) As the time interval gets smaller, the average velocity gets closer to the instantaneous velocity at $t = 1$. We could estimate that the instantaneous velocity at $t = 1$ is $-6.3~cm/s$
