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Math  /  Calculus

QuestionA particle is moving with the given data. Find the position function of the particle. v(t)=sin⁡t−cos⁡t,s(0)=3v(t)=\sin t-\cos t, \quad s(0)=3

Studdy Solution

STEP 1

1. We are given the velocity function v(t)=sin⁡t−cos⁡t v(t) = \sin t - \cos t .
2. We know the initial position s(0)=3 s(0) = 3 .
3. We need to find the position function s(t) s(t) .

STEP 2

1. Integrate the velocity function to find the general form of the position function.
2. Apply the initial condition to determine the constant of integration.
3. Write the complete position function.

STEP 3

Integrate the velocity function v(t)=sin⁡t−cos⁡t v(t) = \sin t - \cos t with respect to t t to find the position function s(t) s(t) :
s(t)=∫(sin⁡t−cos⁡t) dt s(t) = \int (\sin t - \cos t) \, dt

STEP 4

Compute the integral:
s(t)=∫sin⁡t dt−∫cos⁡t dt s(t) = \int \sin t \, dt - \int \cos t \, dt
s(t)=−cos⁡t−sin⁡t+C s(t) = -\cos t - \sin t + C
where C C is the constant of integration.

STEP 5

Use the initial condition s(0)=3 s(0) = 3 to find C C :
s(0)=−cos⁡(0)−sin⁡(0)+C=3 s(0) = -\cos(0) - \sin(0) + C = 3
−1+0+C=3 -1 + 0 + C = 3
C=4 C = 4

STEP 6

Substitute C=4 C = 4 back into the position function:
s(t)=−cos⁡t−sin⁡t+4 s(t) = -\cos t - \sin t + 4
The position function of the particle is:
s(t)=−cos⁡t−sin⁡t+4 s(t) = -\cos t - \sin t + 4

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