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)=4" and "v(0)=10 Which of the following expression gives the velocity of the particle when t=10 ? 10+int_(10)^(0)x(t)dt 10+int_(0)^(10)a(t)dt 10+int_(10)^(0)a(t)dt 10+int_(0)^(10)x(t)dt

Relationship between velocity and acceleration: To find the velocity of the particle at t=10, we need to use the relationship between velocity and acceleration. The velocity at any time t can be found by taking the initial velocity and adding the integral of the acceleration from the initial time to time t.Initial velocity and integral of acceleration: We are given the initial velocity v(0)=10. To find the velocity at t=10, we need to add the integral of the acceleration from time 0 to time 10. This is represented by the expression v(t) = v(0) + ∫(from 0 to t) a(t) dt.Expression for velocity at t=10: The correct expression that represents the velocity of the particle at t=10 is therefore v(10) = 10 + ∫(from 0 to 10) a(t) dt. This corresponds to the second option: 10 + ∫(from 0 to 10) a(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(0)=4" and "v(0)=10
Which of the following expression gives the velocity of the particle when
t=10 ?

10+int_(10)^(0)x(t)dt

10+int_(0)^(10)a(t)dt

10+int_(10)^(0)a(t)dt

10+int_(0)^(10)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: $\newline$ $x(0)=4 \text { and } v(0)=10$ $\newline$ Which of the following expression gives the velocity of the particle when $t=10$ ? $\newline$ $10+\int_{10}^{0} x(t) d t$ $\newline$ $10+\int_{0}^{10} a(t) d t$ $\newline$ $10+\int_{10}^{0} a(t) d t$ $\newline$ $10+\int_{0}^{10} x(t) d t$

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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)=4 \text { and } v(0)=10

\newline

Which of the following expression gives the velocity of the particle when

t=10

\newline

10+\int_{10}^{0} x(t) d t

\newline

10+\int_{0}^{10} a(t) d t

\newline

10+\int_{10}^{0} a(t) d t

\newline

10+\int_{0}^{10} x(t) d t

Relationship between velocity and acceleration: To find the velocity of the particle at $t=10$ , we need to use the relationship between velocity and acceleration. The velocity at any time $t$ can be found by taking the initial velocity and adding the integral of the acceleration from the initial time to time $t$ .
Initial velocity and integral of acceleration: We are given the initial velocity $v(0)=10$ . To find the velocity at $t=10$ , we need to add the integral of the acceleration from time $0$ to time $10$ . This is represented by the expression $v(t) = v(0) + \int_{0}^{t} a(t) \, dt$ .
Expression for velocity at $t=10$ : The correct expression that represents the velocity of the particle at $t=10$ is therefore $v(10) = 10 + \int_{0}^{10} a(t) \, dt$ . This corresponds to the second option: $10 + \int_{0}^{10} a(t) \, dt$ .