Question

Yea thanks thank you for the info thanks

Answer 1

To help us solve this problem let's plot the points given in the table:

From the graph we notice that this the position can be modeled by a sine function, we also notice that the period of this function is 8. We know that a sine function can be modeled by:

`A\sin(B(x+C))+D`

where A is the amplitude, C is the horizontal shift, D is the vertical shift and

`(2\pi)/(B)`

is the period.

From the graph we have we notice that we don't have any horizontal or vertical shift, then C=0 and D=0. We also notice that the amplitude is 15, then A=15. Finally, as we said, the period is 8, then:

`\begin{gathered} 8=(2\pi)/(B) \\ B=(2\pi)/(8) \\ B=(\pi)/(4) \end{gathered}`

Plugging these values in the sine function we have:

`x(t)=15\sin((\pi)/(4)t)`

If we graph this function along the points on the table we get the following graph:

We notice that we don't get an exact fit but we get a close one.

Now, that we have a function that describes the position we can find the velocity by taking the derivative:

`\begin{gathered} x^(\prime)(t)=(d)/(dt)\lbrack15\sin((\pi)/(4)t)\rbrack \\ =(15\pi)/(4)\cos((\pi)/(4)t) \end{gathered}`

Therefore, the velocity is:

`x^(\prime)(t)=(15\pi)/(4)\cos((\pi)/(4)t)`

Once we have the expression for the velocity we can find values for the times we need, they are shown in the table below:

From the table we have that:

`x^(\prime)(0.5)=10.884199\text{ cm/s}`

And that:

• The earliest time when the velocity is zero is 2 s.

,

• The second time when the velocity is zero is 6 s.

,

• The minimum velocity happens at 4 s.

,

• The minimum velocity is -11.780972 cm/s