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(2)=7quad" and "quad v(2)=3 Which of the following expressions gives the displacement of the particle over the interval 2 <= t <= 10 ? int_(2)^(10)|v(t)|dt 7+int_(2)^(10)|v(t)|dt int_(2)^(10)v(t)dt 7+int_(2)^(10)v(t)dt

Integrate Velocity Function: To find the displacement of the particle over the interval from t = 2 to t = 10, we need to integrate the velocity function over this interval. Displacement is the integral of velocity with respect to time.Choose Correct Expression: The given expressions are: 1. ∫_(2)^(10)|v(t)|dt 2. 7+∫_(2)^(10)|v(t)|dt 3. ∫_(2)^(10)v(t)dt 4. 7+∫_(2)^(10)v(t)dt We need to choose the correct expression for displacement. Since displacement can be negative or positive depending on the direction of motion, we do not take the absolute value of the velocity. Therefore, the expressions involving |v(t)| are incorrect.Use Initial Position: We are given the initial position x(2) = 7. This means that the displacement from t = 2 to any other time t is the change in position from x(2), which is the integral of the velocity from t = 2 to that time t plus the initial position x(2).Calculate Displacement: The correct expression for displacement from t = 2 to t = 10 is the initial position at t = 2 plus the integral of the velocity function from t = 2 to t = 10. This is given by the expression: 7 + ∫_(2)^(10)v(t)dt

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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(2)=7quad" and "quad v(2)=3
Which of the following expressions gives the displacement of the particle over the interval
2 <= t <= 10 ?

int_(2)^(10)|v(t)|dt

7+int_(2)^(10)|v(t)|dt

int_(2)^(10)v(t)dt

7+int_(2)^(10)v(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: $\newline$ $x(2)=7 \quad \text { and } \quad v(2)=3$ $\newline$ Which of the following expressions gives the displacement of the particle over the interval $2 \leq t \leq 10$ ? $\newline$ $\int_{2}^{10}|v(t)| d t$ $\newline$ $7+\int_{2}^{10}|v(t)| d t$ $\newline$ $\int_{2}^{10} v(t) d t$ $\newline$ $7+\int_{2}^{10} v(t) d t$

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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(2)=7 \quad \text { and } \quad v(2)=3

\newline

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

2 \leq t \leq 10

\newline

\int_{2}^{10}|v(t)| d t

\newline

7+\int_{2}^{10}|v(t)| d t

\newline

\int_{2}^{10} v(t) d t

\newline

7+\int_{2}^{10} v(t) d t

Integrate Velocity Function: To find the displacement of the particle over the interval from $t = 2$ to $t = 10$ , we need to integrate the velocity function over this interval. Displacement is the integral of velocity with respect to time.
Choose Correct Expression: The given expressions are: $\newline$ $1$ . $\int_{2}^{10}\left|v(t)\right|dt$ $\newline$ $2$ . $7+\int_{2}^{10}\left|v(t)\right|dt$ $\newline$ $3$ . $\int_{2}^{10}v(t)dt$ $\newline$ $4$ . $7+\int_{2}^{10}v(t)dt$ $\newline$ We need to choose the correct expression for displacement. Since displacement can be negative or positive depending on the direction of motion, we do not take the absolute value of the velocity. Therefore, the expressions involving $\left|v(t)\right|$ are incorrect.
Use Initial Position: We are given the initial position $x(2) = 7$ . This means that the displacement from $t = 2$ to any other time $t$ is the change in position from $x(2)$ , which is the integral of the velocity from $t = 2$ to that time $t$ plus the initial position $x(2)$ .
Calculate Displacement: The correct expression for displacement from $t = 2$ to $t = 10$ is the initial position at $t = 2$ plus the integral of the velocity function from $t = 2$ to $t = 10$ . This is given by the expression: $\newline$ $7 + \int_{2}^{10}v(t)\,dt$