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Chapter 4: Problem 93

Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. $$a(t)=3 \sin 2 t ; v(0)=1, s(0)=10$$

Short Answer

Expert verified
Answer: The position function is \(s(t) = -\frac{3}{8} \sin{2t} + \frac{7}{4} t + 10\).

Step by step solution

01

Integrate the acceleration function to find the velocity function

To find the velocity function, integrate the given acceleration function with respect to time: $$v(t) = \int a(t) dt = \int 3 \sin 2t dt$$
02

Evaluate the integral

We can use integration by substitution to solve this integral. Let \(u = 2t\). Then, \(du = 2 dt\). So, the integral becomes $$v(t) = \int \frac{3}{2} \sin{u} du$$ Now, integrate with respect to \(u\) to get: $$v(t) = -\frac{3}{4} \cos{2t} + C_1$$
03

Apply the initial velocity condition

We know that \(v(0) = 1\). Plug in the initial condition to solve for \(C_1\): $$1 = -\frac{3}{4} \cos{0} + C_1$$ Solving for \(C_1\), we get: $$C_1 = \frac{7}{4}$$ Thus, the velocity function becomes: $$v(t) = -\frac{3}{4} \cos{2t} + \frac{7}{4}$$
04

Integrate the velocity function to find the position function

To find the position function, integrate the velocity function with respect to time: $$s(t) = \int v(t) dt = \int \left(-\frac{3}{4} \cos{2t} + \frac{7}{4}\right) dt$$
05

Evaluate the integral

Integrate each term with respect to \(t\): $$s(t) = -\frac{3}{8} \sin{2t} + \frac{7}{4} t + C_2$$
06

Apply the initial position condition

We know that \(s(0) = 10\). Plug in the initial condition to solve for \(C_2\): $$10 = -\frac{3}{8} \sin{0} + \frac{7}{4} (0) + C_2$$ Solving for \(C_2\), we get: $$C_2 = 10$$
07

Write the final position function

Now that we have found \(C_2\), we can write the final position function as: $$s(t) = -\frac{3}{8} \sin{2t} + \frac{7}{4} t + 10$$

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