2. the velocity of a particle moving along the x - axis is given by ( v(t)=sin(2t) ) at time ( t ). if the…

2. the velocity of a particle moving along the x - axis is given by ( v(t)=sin(2t) ) at time ( t ). if the particle is at ( x = 4 ) when ( t = 0 ), what is the position of the particle when ( t=\frac{pi}{2} )?\n(a) 2\n(b) 3\n(c) 4\n(d) 5\n(e) 6

2. the velocity of a particle moving along the x - axis is given by ( v(t)=sin(2t) ) at time ( t ). if the particle is at ( x = 4 ) when ( t = 0 ), what is the position of the particle when ( t=\frac{pi}{2} )?\n(a) 2\n(b) 3\n(c) 4\n(d) 5\n(e) 6

Answer

Explanation:

Step1: Find the position function

We know that (v(t)=\sin(2t)), and (x(t)=\int v(t)dt). Using the integral formula (\int\sin(at)dt =-\frac{1}{a}\cos(at)+C) (here (a = 2)), we have (x(t)=-\frac{1}{2}\cos(2t)+C).

Step2: Determine the constant (C)

Since (x(0) = 4), substitute (t = 0) into (x(t)): (4=-\frac{1}{2}\cos(0)+C). Since (\cos(0)=1), then (4=-\frac{1}{2}+C), so (C = 4+\frac{1}{2}=\frac{9}{2}).

Step3: Calculate (x(\frac{\pi}{2}))

Substitute (t=\frac{\pi}{2}) into (x(t)=-\frac{1}{2}\cos(2t)+\frac{9}{2}). We know that (\cos(2\times\frac{\pi}{2})=\cos(\pi)=- 1). Then (x(\frac{\pi}{2})=-\frac{1}{2}\times(-1)+\frac{9}{2}=\frac{1 + 9}{2}=5).

Answer:

D. 5