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)=7quad" and "quad v(0)=3 Which of the following expressions gives the displacement of the particle over the interval 0 <= t <= 7 ? int_(0)^(7)|x(t)|dt int_(0)^(7)v(t)dt int_(0)^(7)|v(t)|dt int_(0)^(7)x(t)dt

Definition of Displacement: Displacement is defined as the change in position of a particle over a certain time interval. It can be calculated by integrating the velocity function over that time interval.Calculation of Displacement: Given that the position of the particle at time t is x(t) and the velocity is v(t), the displacement from time t = 0 to t = 7 is the integral of the velocity function from 0 to 7.Correct Expression for Displacement: The correct expression for the displacement is therefore the integral of v(t) from 0 to 7, which is written mathematically as: ∫(from 0 to 7) v(t) dtIncorrect Options for Displacement: The other options given are not correct for calculating displacement: - ∫(from 0 to 7) |x(t)| dt would give the total distance traveled without considering direction, which is not the same as displacement. - ∫(from 0 to 7) |v(t)| dt would give the total distance traveled without considering direction, assuming velocity is a scalar. - ∫(from 0 to 7) x(t) dt does not represent displacement, as it would be integrating position over time, which does not yield a meaningful physical quantity in this context.

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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)=7quad" and "quad v(0)=3
Which of the following expressions gives the displacement of the particle over the interval
0 <= t <= 7 ?

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

int_(0)^(7)v(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: $\newline$ $x(0)=7 \quad \text { and } \quad v(0)=3$ $\newline$ Which of the following expressions gives the displacement of the particle over the interval $0 \leq t \leq 7$ ? $\newline$ $\int_{0}^{7}|x(t)| d t$ $\newline$ $\int_{0}^{7} v(t) d t$ $\newline$ $\int_{0}^{7}|v(t)| d t$ $\newline$ $\int_{0}^{7} x(t) d t$

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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)=7 \quad \text { and } \quad v(0)=3

\newline

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

0 \leq t \leq 7

\newline

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

\newline

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

\newline

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

\newline

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

Definition of Displacement: Displacement is defined as the change in position of a particle over a certain time interval. It can be calculated by integrating the velocity function over that time interval.
Calculation of Displacement: Given that the position of the particle at time $t$ is $x(t)$ and the velocity is $v(t)$ , the displacement from time $t = 0$ to $t = 7$ is the integral of the velocity function from $0$ to $7$ .
Correct Expression for Displacement: The correct expression for the displacement is therefore the integral of $v(t)$ from $0$ to $7$ , which is written mathematically as: $\int_{0}^{7} v(t) \, dt$
Incorrect Options for Displacement: The other options given are not correct for calculating displacement: $\newline$ - $\int_{0}^{7} |x(t)| \, dt$ would give the total distance traveled without considering direction, which is not the same as displacement. $\newline$ - $\int_{0}^{7} |v(t)| \, dt$ would give the total distance traveled without considering direction, assuming velocity is a scalar. $\newline$ - $\int_{0}^{7} x(t) \, dt$ does not represent displacement, as it would be integrating position over time, which does not yield a meaningful physical quantity in this context.