for a particle moving along the x - axis, if x(2)=10 and v(2)=8, which of the following expressions gives…

Explanation:

Step1: Recall displacement - velocity relationship

Displacement $\Delta x$ of a particle moving along a line is given by $\Delta x=x(t_2)-x(t_1)=\int_{t_1}^{t_2}v(t)dt$, where $v(t)$ is the velocity - function of the particle and $t_1$ and $t_2$ are the initial and final times respectively. We want to find the displacement of the particle over the interval $2\leq t\leq10$. The displacement $\Delta x$ from $t = 2$ to $t = 10$ is given by the integral of the velocity function over this interval. The initial position $x(2)=10$ is not needed to calculate the displacement over the interval $[2,10]$. The displacement of the particle over the time - interval $[a,b]$ is calculated by integrating the velocity function $v(t)$ from $a$ to $b$. The absolute - value of the velocity function $\vert v(t)\vert$ gives the speed of the particle. Integrating the speed $\int_{a}^{b}\vert v(t)\vert dt$ gives the distance traveled by the particle, not the displacement.

Step2: Identify the correct expression

The displacement of the particle over the interval $2\leq t\leq10$ is given by $\int_{2}^{10}v(t)dt$.

Answer:

$\int_{2}^{10}v(t)dt$