Results 1 to 6 of 6

Math Help - word problem, if your up for a challenge

  1. #1
    Member
    Joined
    Oct 2009
    Posts
    80

    word problem, if your up for a challenge

    A particle moves that its position vector r at the time t is given by

    r = c ( t - 1/3 t^3) i + ct^2 j

    Where c is a positive constant.

    Find the velocity at time t(> 0) and show that’s its magnitude increase with time.

    Determine its direction of motion at time t=0 show that the directions turns through a right angle between t =0 and t=1.

    Show also that the component of force acting on the particle in the direction of the vector j is constant throughout the motion.

    Given that c=1, find the maximum displacement of the particle in the I direction and draw a rough sketch of the path of the particle for values of t> 0, showing clearly the positions of the particle at times t=0, 1, radical 3, 2.
    Last edited by mr fantastic; November 28th 2009 at 01:12 PM. Reason: Changed font size to default so that post could be read without a microscope.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by sderosa518 View Post
    A particle moves that its position vector r at the time t is given by

    r = c ( t - 1/3 t^3) i + ct^2 j

    Where c is a positive constant. Find the velocity at time t(> 0) and show that’s its magnitude increase with time. Determine its direction of motion at time t=0 show that the directions turns through a right angle between t =0 and t=1. Show also that the component of force acting on the particle in the direction of the vector j is constant throughout the motion.
    Given that c=1, find the maximum displacement of the particle in the I direction and draw a rough sketch of the path of the particle for values of t> 0, showing clearly the positions of the particle at times t=0, 1, radical 3, 2.
    Hints for the first couple (if you're up to the challenge of showing all your work and saying where you get stuck):

    v = dr/dt.

    Consider |v| = |dr/dt| and show that d|v|/dt > 0.

    Direction of motion given by direction of dv/dt.

    Take the dot product of dv/dt at t = 0 with dv/dt at t = 1.

    F = ma and a = dv/dt.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Oct 2009
    Posts
    80
    Thank you, but what do you mean find the diravative of what. This question is part of my homework for Calculus Class, plus its my major. I am in college. i was hopin if possible if you can do and explained how to get the answer step by step.

    Thank you
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by sderosa518 View Post
    Thank you, but what do you mean find the diravative of what. This question is part of my homework for Calculus Class, plus its my major. I am in college. i was hopin if possible if you can do and explained how to get the answer step by step.

    Thank you
    You need to differentiate the given vector function. Have you been taught how to do this? If r = f(t) i + g(t) j + h(t) k then dr/dt = df/dt i + dg/dt j + dh/dt k.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Oct 2009
    Posts
    80
    Honestly, my professor isnt that easy going. I am 100% that he didnt taught us this, but we can get help if we need it. Isnt vectors in Calculus II. I am in Cal I. He wants us to think, but nothin is comin to mind. If you can, can you write it out and show me and maybe I can do another example that is in the book.

    TY
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Hello sderosa518
    Quote Originally Posted by sderosa518 View Post
    A particle moves that its position vector r at the time t is given by

    r = c ( t - 1/3 t^3) i + ct^2 j

    Where c is a positive constant.

    Find the velocity at time t(> 0) and show that’s its magnitude increase with time.

    Determine its direction of motion at time t=0 show that the directions turns through a right angle between t =0 and t=1.

    Show also that the component of force acting on the particle in the direction of the vector j is constant throughout the motion.

    Given that c=1, find the maximum displacement of the particle in the I direction and draw a rough sketch of the path of the particle for values of t> 0, showing clearly the positions of the particle at times t=0, 1, radical 3, 2.
    \textbf{r} = c(t-\tfrac13t^3)\textbf{i}+ct^2\textbf{j}

    \Rightarrow \textbf{v}=\frac{d\textbf{r}}{dt}=c(1-t^2)\textbf{i}+2ct\textbf{j}

    The magnitude of the velocity = \sqrt{c^2(1-t^2)^2+4c^2t^2}
    =c\sqrt{1-2t^2+t^4+4t^2}

    =c\sqrt{1+2t^2+t^4}

    =c\sqrt{(1+t^2)^2}

    =c(1+t^2)
    which clearly increases as t increases, for t > 0, since c is a positive constant.

    When t = 0,\; \textbf{v}= c\textbf{i}. So the direction of motion when t=0 is parallel to the vector \textbf{i} (i.e. parallel to the x-axis).

    When t = 1,\; \textbf{v}= 2c\textbf{j}. So the direction of motion when t=1 is parallel to the vector \textbf{j} (i.e. parallel to the y-axis), and is therefore perpendicular to its direction when t = 0.

    Acceleration = \textbf{a} = \frac{d\textbf{v}}{dt} = -2ct\textbf{i}+2c\textbf{j}. The component in the \textbf{j} direction is therefore constant. So the component of the external force that causes this acceleration is also constant in this direction.

    The maximum displacement in the \textbf{i} direction will occur when the component of the velocity in this direction is zero; i.e. when c(1-t^2)=0; i.e. when t = 1 (for t>0)

    So when t = 1 and c = 1, the maximum displacement in the \textbf{i} direction is 1(1-\tfrac13.1^3) = \tfrac23.

    I'll leave you to work out the sketch of the path of the particle. Simply give t various sensible values (including those given in the question) and work out the position vector \textbf{r} for each value. Plot these points to sketch the path.

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Challenge Problem
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: July 12th 2010, 02:07 PM
  2. Challenge Problem
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: September 26th 2009, 01:26 PM
  3. TA’s Challenge Problem #7
    Posted in the Math Challenge Problems Forum
    Replies: 5
    Last Post: August 7th 2009, 08:39 AM
  4. TA’s Challenge Problem #2
    Posted in the Math Challenge Problems Forum
    Replies: 7
    Last Post: June 9th 2009, 02:35 AM
  5. challenge problem
    Posted in the Algebra Forum
    Replies: 6
    Last Post: October 18th 2008, 10:19 PM

Search Tags


/mathhelpforum @mathhelpforum