Question

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

Short answer

Answer: The position function s(t) is given by s(t) = -2cos(t) + t + 2.

Step-by-step solution

Integrate the acceleration function to find the velocity function

Integrate a(t) with respect to time (t): $$ v(t) = \int a(t) \, dt = \int 2 \cos t \, dt $$ Using the integral formula for cosine: $$ v(t) = 2 \sin t + C_1$$ Now, we use the initial condition v(0) = 1: $$ 1 = 2\sin0 + C_1 $$ $$ 1= C_1 $$ So, the velocity function is: $$ v(t) = 2\sin t + 1 $$

Integrate the velocity function to find the position function

Integrate v(t) with respect to time (t): $$ s(t) = \int v(t) \, dt = \int (2\sin t +1) \, dt $$ Using the integral formula for sine and a constant: $$ s(t) = -2\cos t + t + C_2 $$ Now, we use the initial condition s(0) = 0: $$ 0 = -2\cos0 +0 + C_2 $$ $$ 0= -2 + C_2$$ $$ C_2 = 2$$ So, the position function is: $$ s(t) = -2\cos t + t + 2 $$