Question

Find Instantaneous Velocity: The displacement (in centimeters) of a particle moving back and forth along a straight line is givenby the equation s(t) = 2 sin(mt) + 3 cos(mt), where t is measured in seconds.a) Find the average velocity during each time period. Round to 3 decimal places.i) (1, 1.1]ii) [1, 1.01]iii) (1, 1.001]iv) (1, 1.0001] v) [1, 1.00001]b) Estimate the instantaneous velocity of the particle when t = 1.(Hint: b) compare to multiple of Tr]

Answer 1

The average velocity is calculated as:

`(s(t_2)-s(t_1))/(t_2-t_1)`

So, for [1, 1.1], we get that the average velocity is:

`\begin{gathered} s(1)=2\cdot\sin (\pi)+3\cdot\cos (\pi)=-3 \\ s(1.1)=2\cdot\sin (1.1\pi)+3\cdot\cos (1.1\pi)=-3.471 \\ \text{average velocity =}(-3.471-(-3))/(1.1-1)=-4.710 \end{gathered}`

At the same way, For [1, 1.01]:

`\begin{gathered} s(1.01)=-3.061 \\ \text{average velocity = }(-3.061-(-3))/(1.01-1)=-6.100 \end{gathered}`

For [1, 1.001], [1, 1.0001] and for [1.00001]

`\begin{gathered} s(1.001)=-3.006 \\ \text{average velocity = }(-3.006-(-3))/(1.001-1)=-6.000 \end{gathered}`