QuestionFind the position of a particle at time $t=7$ given $a(t)=18t+2$ , $s(0)=16$ , and $v(0)=7$ .

Studdy Solution

STEP 1

Assumptions1. The acceleration of the particle is given by $a(t)=18 t+$ . The initial position of the particle at time $t=0$ is $s(0)=16$
3. The initial velocity of the particle at time $t=0$ is $v(0)=7$
4. We need to find the position of the particle at time $t=7$

STEP 2

First, we need to find the velocity function $v(t)$ of the particle. We can do this by integrating the acceleration function $a(t)$ .
$v(t) = \int a(t) dt$

STEP 3

Now, plug in the given acceleration function into the integral to find the velocity function.
$v(t) = \int (18 t+2) dt$

STEP 4

Calculate the integral to find the velocity function.
$v(t) =9t^2 +2t + C$ Here, $C$ is the constant of integration.

STEP 5

We know that the initial velocity of the particle at time $t=0$ is $v(0)=7$ . We can use this information to find the constant of integration $C$ .
$v(0) =9(0)^2 +2(0) + C =7$

STEP 6

olve the equation for $C$ .
$C =$

STEP 7

Now that we have the constant of integration, we can write the complete velocity function.
$v(t) =9t^2 +2t +7$

STEP 8

Next, we need to find the position function $s(t)$ of the particle. We can do this by integrating the velocity function $v(t)$ .
$s(t) = \int v(t) dt$

STEP 9

Now, plug in the velocity function into the integral to find the position function.
$s(t) = \int (9t^2 +2t +7) dt$

STEP 10

Calculate the integral to find the position function.
$s(t) =3t^3 + t^2 +7t + D$ Here, $$ is the constant of integration.

STEP 11

We know that the initial position of the particle at time $t=0$ is $s(0)=16$ . We can use this information to find the constant of integration $$.
$s(0) =3(0)^3 + (0)^ +7(0) + D =16$

STEP 12

olve the equation for $$.
$=16$

STEP 13

Now that we have the constant of integration, we can write the complete position function.
$s(t) =3t^3 + t^2 +7t +16$

STEP 14

Finally, we can find the position of the particle at time $t=7$ by plugging $t=7$ into the position function.
$s(7) =3(7)^3 + (7)^2 +7(7) +16$

STEP 15

Calculate the position of the particle at time $t=7$ .
$s(7) =3(7)^3 + (7)^2 +7(7) + =1079$ The position of the particle at time $t=7$ is $s(7)=1079$ .

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