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Chapter 12: Problem 6

Given the velocity of an object and its initial position, how do you find the position of the object, for \(t \geq 0 ?\)

Short Answer

Expert verified
Answer: To find the position function, you would first integrate the velocity function with respect to time: \(x(t) = \int v(t) dt\). Then, you would add the initial position, \(x_0\), to the result: \(x(t) = \int v(t) dt + x_0\). This gives the position function, \(x(t)\), for the object as a function of time for \(t \geq 0\).

Step by step solution

01

Write down the given information

The velocity of the object is given as \(v(t)\), and its position at \(t=0\) is given as \(x_0\).
02

Integrate the velocity function

To find the position function, we need to integrate the velocity function with respect to time: \(\int v(t) dt\). Let's call the position function \(x(t)\), so we have: $$x(t) = \int v(t) dt$$
03

Add the initial position

Now we need to account for the initial position of the object, which is given as \(x_0\). We can simply add this constant to our position function: $$x(t) = \int v(t) dt + x_0$$ Now the position function \(x(t)\) represents the position of the object for \(t \geq 0\), taking into account its initial position and velocity.

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