Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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(1)=7quad" and "quad v(1)=4 Which of the following expression gives the position of the particle when t=9? 7+int_(1)^(9)v(t)dt 1+int_(7)^(9)v(t)dt 7+int_(9)^(1)v(t)dt 1+int_(9)^(7)v(t)dt

A particle moves along the x x -axis such that at any time t≥0 t \geq 0 its position is x(t) x(t) , its velocity is v(t) v(t) , and its acceleration is a(t) a(t) . You are given:\newlinex(1)=7 and v(1)=4 x(1)=7 \quad \text { and } \quad v(1)=4 \newlineWhich of the following expression gives the position of the particle when t=9? t=9 ? \newline7+∫19v(t)dt 7+\int_{1}^{9} v(t) d t \newline1+∫79v(t)dt 1+\int_{7}^{9} v(t) d t \newline7+∫91v(t)dt 7+\int_{9}^{1} v(t) d t \newline1+∫97v(t)dt 1+\int_{9}^{7} v(t) d t

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Calculusright chevron icon
Relate position, velocity, speed, and acceleration using derivativesright chevron icon

Full solution

Q. A particle moves along the x x -axis such that at any time t≥0 t \geq 0 its position is x(t) x(t) , its velocity is v(t) v(t) , and its acceleration is a(t) a(t) . You are given:\newlinex(1)=7 and v(1)=4 x(1)=7 \quad \text { and } \quad v(1)=4 \newlineWhich of the following expression gives the position of the particle when t=9? t=9 ? \newline7+∫19v(t)dt 7+\int_{1}^{9} v(t) d t \newline1+∫79v(t)dt 1+\int_{7}^{9} v(t) d t \newline7+∫91v(t)dt 7+\int_{9}^{1} v(t) d t \newline1+∫97v(t)dt 1+\int_{9}^{7} v(t) d t
  1. Understand Question Prompt: The question prompt asks us to find the expression that gives the position of the particle when t=9t=9, given that x(1)=7x(1)=7 and v(1)=4v(1)=4. We know that the position of a particle at any time tt can be found by taking the initial position and adding the integral of the velocity function from the initial time to the time tt.
  2. Initial Position and Value: The correct expression for the position of the particle at time t=9t=9 should start with the initial position at time t=1t=1, which is given as x(1)=7x(1)=7. This means that the initial value in our expression should be 77.
  3. Calculate Integral of Velocity: Next, we need to add the integral of the velocity function v(t)v(t) from the initial time t=1t=1 to the final time t=9t=9. This is represented mathematically as the integral from 11 to 99 of v(t) dtv(t) \, dt. This integral will give us the change in position from time t=1t=1 to time t=9t=9.
  4. Final Expression for Position: Therefore, the correct expression that gives the position of the particle when t=9t=9 is the initial position plus the integral of the velocity function from time t=1t=1 to t=9t=9. This is represented as:\newline7+∫19v(t) dt7 + \int_{1}^{9} v(t) \, dt.
  5. Match with Provided Options: Looking at the options provided, the expression that matches our derived expression is: 7+∫19v(t) dt7 + \int_{1}^{9} v(t) \, dt.

More problems from Relate position, velocity, speed, and acceleration using derivatives

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t0.7sin⁡(2t+3) v(t)=t^{0.7} \sin (2 t+3) . What is the acceleration of the particle at time t=4 t=4 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t1.1cos⁡(t2−5) v(t)=t^{1.1} \cos \left(t^{2}-5\right) . If the position of the particle is x=−2 x=-2 when t=2.5 t=2.5 , what is the position of the particle when t=1 t=1 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t0.9sin⁡(t−1) v(t)=t^{0.9} \sin (t-1) . If the position of the particle is x=−3 x=-3 when t=3 t=3 , what is the position of the particle when t=5 t=5 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t2.1sin⁡(2t) v(t)=t^{2.1} \sin (2 t) . Find all times when the speed of the particle is equal to 22 on the interval 0≤t≤4 0 \leq t \leq 4 . You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer: t= t=
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=(t2.3+5)sin⁡(2t) v(t)=\left(t^{2.3}+5\right) \sin (2 t) . What is the distance traveled by the particle over the interval 0≤t≤5 0 \leq t \leq 5 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t1.9cos⁡(3t−1) v(t)=t^{1.9} \cos (3 t-1) . What is the distance traveled by the particle over the interval 0≤t≤3 0 \leq t \leq 3 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=(t1.6+3)cos⁡(3t) v(t)=\left(t^{1.6}+3\right) \cos (3 t) . What is the average velocity of the particle on the interval 0≤t≤6 0 \leq t \leq 6 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its acceleration is given by a(t)=(t0.5+3)cos⁡(3t) a(t)=\left(t^{0.5}+3\right) \cos (3 t) . If the velocity of the particle is v=−2 v=-2 when t=1 t=1 , what is the velocity of the particle when t=1 t=1 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=t2.1sin⁡(2t+5) v(t)=t^{2.1} \sin (2 t+5) . Find all times when the speed of the particle is equal to 11 on the interval 0≤t≤3 0 \leq t \leq 3 . You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer: t= t=
Get tutor helpright-arrow

Posted 9 months ago

Question
A particle travels along the x x -axis such that its velocity is given by v(t)=(t2.1+5)cos⁡(2t) v(t)=\left(t^{2.1}+5\right) \cos (2 t) . What is the distance traveled by the particle over the interval 0≤t≤5 0 \leq t \leq 5 ? You may use a calculator and round your answer to the nearest thousandth.\newlineAnswer:
Get tutor helpright-arrow

Posted 9 months ago

View all (107)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant