A particle moves along the x-axis such that at any time t >= 0 its position is x(t), its velocity is v(t), and its acceleration is a(t). You are given: x(0)=5quad" and "quad v(0)=6 Which of the following expressions gives the displacement of the particle over the interval 0 <= t <= 7 ? int_(0)^(7)|v(t)|dt int_(0)^(7)x(t)dt int_(0)^(7)v(t)dt int_(0)^(7)|x(t)|dt

Define Displacement Integration: To find the displacement of the particle over a given time interval, we need to integrate the velocity function over that interval. Displacement is the integral of velocity with respect to time.Eliminate Irrelevant Options: The given options are: 1. ∫(0 to 7)|v(t)|dt 2. ∫(0 to 7)x(t)dt 3. ∫(0 to 7)v(t)dt 4. ∫(0 to 7)|x(t)|dt We can eliminate options 1 and 4 because the absolute value of velocity or position is not relevant to displacement; it would be relevant to distance traveled, which is not what is being asked.Identify Correct Expression: Option 2, ∫(0 to 7)x(t)dt, is also incorrect because integrating the position function over time does not give displacement. It would give the accumulated position, which is not a standard concept in kinematics.Option 3, ∫(0 to 7)v(t)dt, is the correct expression for displacement. It integrates the velocity function over the time interval from 0 to 7, which gives the total change in position, or displacement, over that interval.

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A particle moves along the
x-axis such that at any time
t >= 0 its position is
x(t), its velocity is
v(t), and its acceleration is
a(t). You are given:

x(0)=5quad" and "quad v(0)=6
Which of the following expressions gives the displacement of the particle over the interval
0 <= t <= 7 ?

int_(0)^(7)|v(t)|dt

int_(0)^(7)x(t)dt

int_(0)^(7)v(t)dt

int_(0)^(7)|x(t)|dt

A particle moves along the $x$ -axis such that at any time $t \geq 0$ its position is $x(t)$ , its velocity is $v(t)$ , and its acceleration is $a(t)$ . You are given: $\newline$ $x(0)=5 \quad \text { and } \quad v(0)=6$ $\newline$ Which of the following expressions gives the displacement of the particle over the interval $0 \leq t \leq 7$ ? $\newline$ $\int_{0}^{7}|v(t)| d t$ $\newline$ $\int_{0}^{7} x(t) d t$ $\newline$ $\int_{0}^{7} v(t) d t$ $\newline$ $\int_{0}^{7}|x(t)| d t$

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Math Problems

Calculus

Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle moves along the

x

-axis such that at any time

t \geq 0

its position is

x(t)

, its velocity is

v(t)

, and its acceleration is

a(t)

. You are given:

\newline

x(0)=5 \quad \text { and } \quad v(0)=6

\newline

Which of the following expressions gives the displacement of the particle over the interval

0 \leq t \leq 7

\newline

\int_{0}^{7}|v(t)| d t

\newline

\int_{0}^{7} x(t) d t

\newline

\int_{0}^{7} v(t) d t

\newline

\int_{0}^{7}|x(t)| d t

Define Displacement Integration: To find the displacement of the particle over a given time interval, we need to integrate the velocity function over that interval. Displacement is the integral of velocity with respect to time, represented as $\int v(t) \, dt$ .
Eliminate Irrelevant Options: The given options are: $\newline$ $1$ . $\int_{0}^{7}|v(t)|\,dt$ $\newline$ $2$ . $\int_{0}^{7}x(t)\,dt$ $\newline$ $3$ . $\int_{0}^{7}v(t)\,dt$ $\newline$ $4$ . $\int_{0}^{7}|x(t)|\,dt$ $\newline$ We can eliminate options $1$ and $4$ because the absolute value of velocity or position is not relevant to displacement; it would be relevant to distance traveled, which is not what is being asked.
Identify Correct Expression: Option $2$ , $\int_{0}^{7}x(t)dt$ , is also incorrect because integrating the position function over time does not give displacement. It would give the accumulated position, which is not a standard concept in kinematics.
Identify Correct Expression: Option $2$ , $\int_{0}^{7}x(t)dt$ , is also incorrect because integrating the position function over time does not give displacement. It would give the accumulated position, which is not a standard concept in kinematics.Option $3$ , $\int_{0}^{7}v(t)dt$ , is the correct expression for displacement. It integrates the velocity function over the time interval from $0$ to $7$ , which gives the total change in position, or displacement, over that interval.