A mass attached to a vertical spring has position function given by where is measured in seconds and in inches. Find the velocity at time . in/s Find the acceleration at time . in/s2

Question

A mass attached to a vertical spring has position function given by 𝑠(𝑡)=4sin(3𝑡) where 𝑡 is measured in seconds and 𝑠 in inches.

Find the velocity at time 𝑡=4.    in/s

Find the acceleration at time 𝑡=4.    in/s²

Answer 1

Hello Novalee

The instantaneous velocity of the mass is the derivative of the position function [with respect to time.]

So

v(t) = s'(t)

v(t) = 4cos(3t)⋅3 (using the chain rule.)

v(t) = 12cos(3t)

Now substitute to obtain the velocity at t=4.

v(4) = 12cos(3⋅4)

v(4) = 12cos(12)

v(4) = 10.126 in/s (to three decimal places.)

The instantaneous acceleration of the mass is the derivative of the velocity function [with respect to time.]

a(t) = v'(t)

a(t) = -12sin(3t)⋅3

a(t) = -36sin(3t)

Again substitute t = 4.

a(t) = -36sin(3⋅4) = -19.317 in/s².

Hope this helps. Let me know if you have any questions.

William