The function $a\left(t\right)=4-\sin \left(t\right)$ describes the acceleration of a particle. At $t=0$, the particle is at $x=-20m$ and moving with a velocity of $+2\text{ m/s}$. What is the particle's position at $t=8s$?

• A. $x=117m$
• B. $x=101m$
• C. $x=245m$
• D. $x=115m$

Answer: (A)

Verified step by step guidance

1. Identify the given acceleration function: \(a(t) = 4 - \sin(t)\), and note the initial conditions: at \(t=0\), position \(x(0) = -20\) m and velocity \(v(0) = +2\) m/s.
2. Recall that acceleration is the derivative of velocity with respect to time, so to find velocity as a function of time, integrate the acceleration: \(v(t) = \int a(t) \, dt = \int (4 - \sin(t)) \, dt\).
3. Perform the integration to find \(v(t)\), remembering to include the constant of integration \(C_1\), which can be found using the initial velocity condition \(v(0) = 2\) m/s.
4. Next, find the position function \(x(t)\) by integrating the velocity function: \(x(t) = \int v(t) \, dt\), again including a constant of integration \(C_2\) determined by the initial position \(x(0) = -20\) m.
5. Finally, evaluate the position function \(x(t)\) at \(t=8\) seconds to find the particle's position at that time.