Question

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=-32;v(0)=20,s(0)=0$$

Short answer

Question: Determine the position function of an object moving along a line with an acceleration function of $a(t)=-32$, an initial velocity of $v(0)=20$, and an initial position of $s(0)=0$. Answer: The position function of the object is $s(t)=-16t^2+20t$.

Step-by-step solution

Integrate the acceleration function to find the velocity function

Integrate the acceleration function, $a(t)=-32$, with respect to time: $$v(t)=\int a(t)dt=\int (-32)dt=-32t+C_1$$ We are given the initial velocity: $v(0)=20$. Use this condition to find the constant $C_1$: $$v(0)=-32(0)+C_1\Rightarrow 20=C_1$$ So the velocity function is: $$v(t)=-32t+20$$

Integrate the velocity function to find the position function

Integrate the velocity function, $v(t)=-32t+20$, with respect to time: $$s(t)=\int v(t)dt=\int (-32t+20)dt=-16t^2+20t+C_2$$ We are given the initial position: $s(0)=0$. Use this condition to find the constant $C_2$: $$s(0)=-16(0{)}^2+20(0)+C_2\Rightarrow 0=C_2$$ So the position function is: $$s(t)=-16t^2+20t$$ The position function of the object moving along a line is: $$s(t)=-16t^2+20t$$