Question

A particle is moving with acceleration a ( t ) = 18 t + 16 . Its position at time t = 0 is s ( 0 ) = 10 and its velocity at time t = 0 is v ( 0 ) = 16 . What is its position at time t = 8 ?

Answer 1

`s(8)=2186`

Explanation:

A particle is moving with acceleration modeled by the function:

`a(t)=18t+16`

We are given that its position s(t) at t = 0 is 10 and that its velocity v(t) at t = 0 is 16.

And we want to find its position at t = 8.

Velocity is the integral of the acceleration. Hence:

`\displaystyle v(t)=\int a(t)\, dt=\int 18t+16\, dt`

Find the velocity. Remember the constant of integration!

`v(t)=9t^2+16t+C`

Since v(t) is 16 when t = 0:

`(16)=9(0)^2+16(0)+C\Rightarrow C=16`

Hence, our velocity is given by:

`v(t)=9t^2+16t+16`

Position is the integral of the velocity. Hence:

`\displaystyle s(t)=\int v(t)\, dt=\int 9t^2+16t+16\, dt`

Integrate:

`\displaystyle s(t)=3t^3+8t^2+16t+C`

s(t) is 10 when t = 10. Hence:

`C=10`

So, our position function is:

`s(t)=3t^3+8t^2+16t+10`

The position at t = 8 will be:

`s(8)=3(8)^3+8(8)^2+16(8)+10=2186`