record: 4 score: 4 for a particle moving along the x - axis, if x(2)=5 and v(2)=7, which of the following…

record: 4 score: 4 for a particle moving along the x - axis, if x(2)=5 and v(2)=7, which of the following expressions gives the displacement of the particle over the interval 2 ≤ t ≤ 8? ∫₂⁸|v(t)|dt 5 + ∫₂⁸v(t)dt ∫₂⁸v(t)dt 5 + ∫₂⁸|v(t)|dt

record: 4 score: 4 for a particle moving along the x - axis, if x(2)=5 and v(2)=7, which of the following expressions gives the displacement of the particle over the interval 2 ≤ t ≤ 8? ∫₂⁸|v(t)|dt 5 + ∫₂⁸v(t)dt ∫₂⁸v(t)dt 5 + ∫₂⁸|v(t)|dt

Answer

Explanation:

Step1: Recall displacement - velocity relationship

Displacement $\Delta x$ over an interval $[a,b]$ is given by $\Delta x=x(b)-x(a)$. Also, the velocity - position relationship is $v(t)=\frac{dx}{dt}$, so $x(t)=x(a)+\int_{a}^{t}v(s)ds$.

Step2: Identify initial conditions

We are given $x(2) = 5$ and we want to find the displacement over the interval $[2,8]$. The displacement $\Delta x=x(8)-x(2)$.

Step3: Express $x(8)$ in terms of integral

Since $x(t)=x(2)+\int_{2}^{t}v(s)ds$, then $x(8)=x(2)+\int_{2}^{8}v(s)ds$. And the displacement $\Delta x=x(8)-x(2)=\int_{2}^{8}v(s)ds$. The absolute - value of velocity $|v(t)|$ gives the speed, and integrating speed gives the distance traveled, not the displacement.

Answer:

$\int_{2}^{8}v(t)dt$