To determine the position function [tex]\( s(t) \)[/tex], given the acceleration function [tex]\( a(t) = -28 \)[/tex], initial velocity [tex]\( v(0) = 23 \)[/tex], and initial position [tex]\( s(0) = 0 \)[/tex], we will proceed through the following steps:
1. Integrate the acceleration function to find the velocity function:
The acceleration function is given as:
[tex]\[
a(t) = -28
\][/tex]
To find the velocity function [tex]\( v(t) \)[/tex], we integrate the acceleration function with respect to time [tex]\( t \)[/tex]:
[tex]\[
v(t) = \int a(t) \, dt = \int -28 \, dt = -28t + C_1
\][/tex]
Here, [tex]\( C_1 \)[/tex] is the constant of integration. To determine [tex]\( C_1 \)[/tex], we use the initial condition for velocity [tex]\( v(0) = 23 \)[/tex]:
[tex]\[
v(0) = -28(0) + C_1 = 23 \implies C_1 = 23
\][/tex]
Therefore, the velocity function is:
[tex]\[
v(t) = -28t + 23
\][/tex]
2. Integrate the velocity function to find the position function:
With the velocity function known, we integrate it to find the position function [tex]\( s(t) \)[/tex]:
[tex]\[
s(t) = \int v(t) \, dt = \int (-28t + 23) \, dt
\][/tex]
We will integrate each term separately:
[tex]\[
s(t) = \int -28t \, dt + \int 23 \, dt = -14t^2 + 23t + C_2
\][/tex]
Here, [tex]\( C_2 \)[/tex] is another constant of integration. To determine [tex]\( C_2 \)[/tex], we use the initial condition for position [tex]\( s(0) = 0 \)[/tex]:
[tex]\[
s(0) = -14(0)^2 + 23(0) + C_2 = 0 \implies C_2 = 0
\][/tex]
Therefore, the position function is:
[tex]\[
s(t) = -14t^2 + 23t
\][/tex]
Thus, the position function [tex]\( s(t) \)[/tex] is given by:
[tex]\[
s(t) = -14t^2 + 23t
\][/tex]
