Chapter 5, Problem 5RCC

(a)

To determine

To define: The integral ${\displaystyle {\int }_{60}^{120}v\left(t\right)dt}$.

Explanation of Solution

Given information:

The velocity of the particle over time is v(t). It is measured in feet per second.

The acceleration of the particle is $a\left(t\right)$.

The particle is moves back and forth along the straight line.

The upper limit is 120 seconds (2 minute) and lower limit is 60 seconds (1 minute).

The displacement is an integral of velocity (v) over time (t).

Show the integral function as follows:

${\displaystyle {\int }_{60}^{120}v\left(t\right)dt}$ (1)

Equation (1) represents the displacement of the particle from first to second minute duration.

Therefore, the displacement of the particle from first to second minute time duration is represented as ${\displaystyle {\int }_{60}^{120}v\left(t\right)dt}$.

(b)

To determine

To define: The integral ${\displaystyle {\int }_{60}^{120}\left|v\left(t\right)\right|dt}$.

Explanation of Solution

Show the integral function as follows:

${\displaystyle {\int }_{60}^{120}v\left(t\right)dt}$ (2)

Equation (2) represents the total distance travelled by the particle from first to second minute duration.

The sum of the distance traveled by the particle in the straight line on time interval of $t=60$ to $t=120\text{seconds}$.

Therefore ${\displaystyle {\int }_{60}^{120}\left|v\left(t\right)\right|dt}$ represents the total distance of the particle moves from first to second minute along the straight line.

(c)

To determine

To define: The integral ${\displaystyle {\int }_{60}^{120}a\left(t\right)dt}$.

Explanation of Solution

The velocity is an integral of acceleration (a) over time (t).

Show the integral function as follows:

${\displaystyle {\int }_{60}^{120}a\left(t\right)dt}$ (3)

Equation (3) represents the velocity change of the particle from first to second minute duration.

Therefore ${\displaystyle {\int }_{60}^{120}a\left(t\right)dt}$ represents the rate of change in the velocity of particle on time interval of $t=60$ to $t=120\text{seconds}$.