Problem 47 - Calculus: Early Transcendental Functions

Question

Find the position function \(s(t)\) from the given velocity or acceleration function and initial value(s). Assume that units are feet and seconds. $$v(t)=40-\sin t, s(0)=2$$

Step-by-Step Solution

Answer

The position function is \(s(t) = 40t - \cos(t) + 3\).

Step 1: Understand the relationship between velocity and position

In physics, velocity is the derivative of the position with respect to time. Therefore, the position is the integral of the velocity.

Step 2: Calculate the anti-derivative of the velocity function

To find the position function, the anti-derivative of the velocity function \(v(t) = 40 - \sin(t)\) should be calculated. The anti-derivative of \(40 - \sin(t)\) can be calculated as \(40t - \cos(t)\).

Step 3: Add the constant of integration

After evaluation, a constant of integration C should be added. The antiderivative actually represents a family of functions, as integrating loses the constant term. To find the exact function that describes the position, the constant of integration needs to be determined using the initial conditions given.

Step 4: Determine the constant of integration

The initial condition that \(s(0) = 2\) is given. Substituting these in the equation obtained, resulting in \(2 = 40 * 0 - \cos(0) + C\) which simplifies to \(2 = -1 + C\). Hence, the constant of integration C is 3.

Step 5: Write down the position function

Finally, substitute the constant into the integral gives the position function: \(s(t) = 40t - \cos(t) + 3\).