Question

A spring hanging from the ceiling vibrates up and down. it's vertical position at time (t) is given by f(t)=4 sin(3t). (a) Find the velocity of the spring at time t. (b) What is the spring's maximum speed. (c) what is location when it reaches its maximum speed.

Answer 1

You know that the position as a function of time is:
\[x(t) = 4\sin(3t)\]
you also know that velocity is defined as:
\[\frac{dx}{dt}\]

Answer 2

can you find the derivative of:
\[x(t) = 4\sin(3t)\]

Answer 3

Yes that is the ans of a.what about b and c parts?

Answer 4

so we want to find the maximum of:
\[12\cos(3t)\]

Answer 5

So this basically comes down to finding where the cosine function has it's maximums

Answer 6

and then using the value for the maximum of a cosine in the equation

Answer 7

So, what is the maximum of a cosine function?

Answer 8

1

Answer 9

yes, so the maximum value for the speed must be:

Answer 10

i think 12

Answer 11

yes

Answer 12

At what time does the speed become 12?

Answer 13

don't know.may be 0

Answer 14

Yes, because cos(x) = 1 when x = 0

Answer 15

Now that you know the time at which it reaches its maximum speed, you can plug that into the first equation

Answer 16

which first equation?

Answer 17

the equation for position as a function of time

Answer 18

f(t)=4sin(3t)

Answer 19

yes

Answer 20

what i do with this now? please explain

Answer 21

i put t=0 so position is zero?

Answer 22

Yes

Answer 23

ok.Thanks a lot.

Answer 24

You are welcome