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Chapter 4: Problem 91

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=0.2 t ; v(0)=0, s(0)=1$$

Short Answer

Expert verified
Answer: The position function of the object is $$s(t) = \frac{1}{30}t^3 + 1$$.

Step by step solution

01

Integrate the acceleration function to obtain the velocity function

Integrate the acceleration function, $$a(t)=0.2t$$ with respect to time (t) to find the velocity function: $$v(t) = \int a(t) dt = \int 0.2t dt$$ This integration gives us: $$v(t) = 0.1t^2 + C_1$$ where $$C_1$$ is the constant of integration.
02

Use the initial velocity condition to find the constant of integration$$C_1$$

Since the initial velocity is given as $$v(0)=0$$, we can use this information to solve for the constant $$C_1$$: $$0 = 0.1(0)^2 + C_1$$ From this equation, we can see that $$C_1 = 0$$.
03

Write the obtained velocity function

Now that we have found the constant $$C_1$$, we can write the velocity function as: $$v(t) = 0.1t^2$$
04

Integrate the velocity function to obtain the position function

Next, integrate the velocity function, $$v(t)=0.1t^2$$ with respect to time (t) to find the position function: $$s(t) = \int v(t) dt = \int 0.1t^2 dt$$ This integration gives us: $$s(t) = \frac{1}{30}t^3 + C_2$$ where $$C_2$$ is the constant of integration.
05

Use the initial position condition to find the constant of integration$$C_2$$

Since the initial position is given as $$s(0)=1$$, we can use this information to solve for the constant $$C_2$$: $$1 = \frac{1}{30}(0)^3 + C_2$$ From this equation, we can see that $$C_2 = 1$$.
06

Write the obtained position function

Now that we have found the constant $$C_2$$, we can write the position function as: $$s(t) = \frac{1}{30}t^3 + 1$$ The position function of the object moving along a line is: $$s(t) = \frac{1}{30}t^3 + 1$$

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Most popular questions from this chapter

The velocity function and initial position of Runners \(A\) and \(B\) are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other. $$\text { A: } v(t)=2 e^{-t}, s(0)=0 ; \quad \text { B: } V(t)=4 e^{-4 t}, S(0)=10$$

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