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Finding the magnitude of displacement with a given parametric velocity function

position question from velocity and given point.Find parametric equationsProblem with moving particle along a straight line.Velocity of a curve given by parametric equationsCalculate the average acceleration and average speed of a particleFinding average velocity given curved graph?Calculus: Does the particle ever change direction? At what time intervals does velocity increase?Find the total distance traveled by a particle between $t = 1$ and $t = 3$Velocity/Displacement questionFinding the displacement and distance using integral as net change concept.

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I am struggling to get the correct answer for the question: what is the magnitude of the displacement of a particle moving in thee xy-plane with the velocity vector given by $v(t) = <e^left(sintright), 5t^2>$ for time $t ge 0$ between time $t=1$ and $t=2$. I attempted to take the integral of the speed function $int_1^2 (sqrte^left(2sintright)+25t^4) , textrmdt \$ since the velocity of the particle is always positive on the interval and got the answer $11.992$. However, apparently the correct answer is $11.954$, but I can't find a way to come about that answer.

calculus integration derivatives parametric

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asked Mar 16 at 23:38

LudwigLudwig

875

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  What you need is $ left| int_1^2 v mathrmdtright| $, not $ int_1^2left| v right|mathrmdt$.
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  – ChargeShivers
  Mar 16 at 23:57
  
  

  
  
  
  


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  The total distance travelled by a particle obeying the velocity function $v(t)$ is $int_a^b |v(t)| dt$. The displacement of a particle under the same law is $int_b^av (t) dt$. If you want the magnitude of the displacement, calculate the latter integral and take the absolute value.
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  – coreyman317
  Mar 16 at 23:57
  
  

  
  
  
  


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  @coreyman317 how would I go about evaluating that integral since $v(t)$ is a parametric function?
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  – Ludwig
  Mar 17 at 4:11
  

  
  
  
  


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  @ChargeShivers how would I go about evaluating that integral since v(t) is a parametric function?
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  – Ludwig
  Mar 17 at 4:27
  

  
  
  
  


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  Let $v = (v_x, v_y)$, then $int v mathrmdt = left( int v_x mathrmdt, int v_ymathrmdtright)$, and so $left| int v mathrmdt right| = sqrt left( int v_x mathrmdt right)^2 + left( int v_y mathrmdt right)^2 $.
  $endgroup$
  – ChargeShivers
  Mar 18 at 3:39
  

  
  
  
  

add a comment | 

0

$begingroup$

I am struggling to get the correct answer for the question: what is the magnitude of the displacement of a particle moving in thee xy-plane with the velocity vector given by $v(t) = <e^left(sintright), 5t^2>$ for time $t ge 0$ between time $t=1$ and $t=2$. I attempted to take the integral of the speed function $int_1^2 (sqrte^left(2sintright)+25t^4) , textrmdt \$ since the velocity of the particle is always positive on the interval and got the answer $11.992$. However, apparently the correct answer is $11.954$, but I can't find a way to come about that answer.

calculus integration derivatives parametric

share|cite|improve this question

asked Mar 16 at 23:38

LudwigLudwig

875

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What you need is $ left| int_1^2 v mathrmdtright| $, not $ int_1^2left| v right|mathrmdt$.
  $endgroup$
  – ChargeShivers
  Mar 16 at 23:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The total distance travelled by a particle obeying the velocity function $v(t)$ is $int_a^b |v(t)| dt$. The displacement of a particle under the same law is $int_b^av (t) dt$. If you want the magnitude of the displacement, calculate the latter integral and take the absolute value.
  $endgroup$
  – coreyman317
  Mar 16 at 23:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @coreyman317 how would I go about evaluating that integral since $v(t)$ is a parametric function?
  $endgroup$
  – Ludwig
  Mar 17 at 4:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @ChargeShivers how would I go about evaluating that integral since v(t) is a parametric function?
  $endgroup$
  – Ludwig
  Mar 17 at 4:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Let $v = (v_x, v_y)$, then $int v mathrmdt = left( int v_x mathrmdt, int v_ymathrmdtright)$, and so $left| int v mathrmdt right| = sqrt left( int v_x mathrmdt right)^2 + left( int v_y mathrmdt right)^2 $.
  $endgroup$
  – ChargeShivers
  Mar 18 at 3:39
  

  
  
  
  

add a comment | 

$begingroup$

I am struggling to get the correct answer for the question: what is the magnitude of the displacement of a particle moving in thee xy-plane with the velocity vector given by $v(t) = <e^left(sintright), 5t^2>$ for time $t ge 0$ between time $t=1$ and $t=2$. I attempted to take the integral of the speed function $int_1^2 (sqrte^left(2sintright)+25t^4) , textrmdt \$ since the velocity of the particle is always positive on the interval and got the answer $11.992$. However, apparently the correct answer is $11.954$, but I can't find a way to come about that answer.

calculus integration derivatives parametric

share|cite|improve this question

asked Mar 16 at 23:38

LudwigLudwig

875

$endgroup$

share|cite|improve this question

asked Mar 16 at 23:38

LudwigLudwig

875

share|cite|improve this question

asked Mar 16 at 23:38

LudwigLudwig

875

asked Mar 16 at 23:38

LudwigLudwig

875

asked Mar 16 at 23:38

LudwigLudwig

875