question 3\na particle is moving with the given data. find the position of the particle.\n$v(t)=\\sin t…

question 3\na particle is moving with the given data. find the position of the particle.\n$v(t)=\\sin t - \\cos t$, $s(0)=4$\n$\\bigcirc s(t)=5 - \\cos t - \\sin t$\n$\\bigcirc s(t)=1 - t - \\sin t$\n$\\bigcirc s(t)=\\cos^2 t$\n$\\bigcirc s(t)=\\cos t - \\sin t$\n$\\bigcirc s(t)=1 - \\cos t + \\sin t$

question 3\na particle is moving with the given data. find the position of the particle.\n$v(t)=\\sin t - \\cos t$, $s(0)=4$\n$\\bigcirc s(t)=5 - \\cos t - \\sin t$\n$\\bigcirc s(t)=1 - t - \\sin t$\n$\\bigcirc s(t)=\\cos^2 t$\n$\\bigcirc s(t)=\\cos t - \\sin t$\n$\\bigcirc s(t)=1 - \\cos t + \\sin t$

Answer

Explanation:

Step1: Integrate velocity function

Since (v(t)=\sin t-\cos t), and (s(t)=\int v(t)dt). Using (\int\sin tdt = -\cos t + C_1) and (\int\cos tdt=\sin t + C_2), we have (s(t)=-\cos t-\sin t + C).

Step2: Use initial condition

Given (s(0) = 4). Substitute (t = 0) into (s(t)): (s(0)=-\cos(0)-\sin(0)+C). Since (\cos(0) = 1) and (\sin(0)=0), then (4=-1 - 0+C). Solving for (C), we get (C = 5).

Answer:

(s(t)=5-\cos t-\sin t)