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Thread: Inequality

  1. #1
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    Inequality

    Q:
    Solve $\displaystyle \frac{2}{5} (2-3e)u < 0 $ to find a range for $\displaystyle e$.

    My Problem:
    Is it $\displaystyle e> \frac{2}{3}$ or $\displaystyle e < \frac{2}{3}$. I'm getting the answer as $\displaystyle e < \frac{2}{3}$ but the mark scheme says that it is $\displaystyle e> \frac{2}{3}$. Can someone confirm the correct answer? Thanks in advance.
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    Quote Originally Posted by Air View Post
    Q:
    Solve $\displaystyle \frac{2}{5} (2-3e)u < 0 $ to find a range for $\displaystyle e$.

    My Problem:
    Is it $\displaystyle e> \frac{2}{3}$ or $\displaystyle e < \frac{2}{3}$. I'm getting the answer as $\displaystyle e < \frac{2}{3}$ but the mark scheme says that it is $\displaystyle e> \frac{2}{3}$. Can someone confirm the correct answer? Thanks in advance.
    Hey Air,

    What is u? If the answer says so, then mostly u < 0
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    Quote Originally Posted by Isomorphism View Post
    Hey Air,

    What is u? If the answer says so, then mostly u < 0
    $\displaystyle u$ doesn't have a value so it can be divided by both sides to get rid of it. The question is asking to find the range coefficient of restitution ($\displaystyle e$).

    [It's a follow-on part of my method]
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    Quote Originally Posted by Air View Post
    $\displaystyle u$ doesn't have a value so it can be divided by both sides to get rid of it. The question is asking to find the range coefficient of restitution ($\displaystyle e$).

    [It's a follow-on part of my method]
    Oh that means the velocity u was negative... If u is negative then while dividing both sides by it, the inequality toggles.Or see it this way:

    Product of two numbers is positive if both are positive or both are negative. Mostly your velocity u is negative and thus 2 - 3e must also be negative.
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    Quote Originally Posted by Isomorphism View Post
    Oh that means the velocity u was negative... If u is negative then while dividing both sides by it, the inequality toggles.Or see it this way:

    Product of two numbers is positive if both are positive or both are negative. Mostly your velocity u is negative and thus 2 - 3e must also be negative.
    So, which is the correct inequality: $\displaystyle e < \frac{2}{3}$ or $\displaystyle e > \frac{2}{3}$?
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    Quote Originally Posted by Air View Post
    So, which is the correct inequality: $\displaystyle e < \frac{2}{3}$ or $\displaystyle e > \frac{2}{3}$?
    Hmm again could you just post the problem entirely.... Or you could confirm the sign of u. Then I will be able to tell.
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    Quote Originally Posted by Isomorphism View Post
    Hmm again could you just post the problem entirely.... Or you could confirm the sign of u. Then I will be able to tell.

    Q:

    A particle $\displaystyle P$ of mass $\displaystyle 3m$ is moving with speed $\displaystyle 2u$ in a straight line on a smooth horizontal table. The particle $\displaystyle P$ collides with a particle $\displaystyle Q$ of mass $\displaystyle 2m$ moving with speed $\displaystyle u$ in the opposite direction to $\displaystyle P$.

    a) Show that the speed of Q after the collisions is $\displaystyle \frac15 u (9e +4)$.

    As a result of the collision, the direction of motion of $\displaystyle P$ is reversed.

    (b) Find the range of possible values of $\displaystyle e$.

    [It's part b that we are discussing, which a follow on question]
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    Quote Originally Posted by Air View Post

    Q:

    A particle $\displaystyle P$ of mass $\displaystyle 3m$ is moving with speed $\displaystyle 2u$ in a straight line on a smooth horizontal table. The particle $\displaystyle P$ collides with a particle $\displaystyle Q$ of mass $\displaystyle 2m$ moving with speed $\displaystyle u$ in the opposite direction to $\displaystyle P$.

    a) Show that the speed of Q after the collisions is $\displaystyle \frac15 u (9e +4)$.

    As a result of the collision, the direction of motion of $\displaystyle P$ is reversed.

    (b) Find the range of possible values of $\displaystyle e$.

    [It's part b that we are discussing, which a follow on question]
    Ya so I get $\displaystyle v_{P} = \frac{4u - 6ue}5 < 0$ with u positive. This means $\displaystyle 4 - 6e < 0 \Rightarrow 2 < 3e \Rightarrow e > \frac23$
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    Thanks Isomorphism!

    Also, there is another follow-on question, which is:

    c) Given that the magnitude of the impulse of $\displaystyle P$ on $\displaystyle Q$ is $\displaystyle \frac{32}{5} mu$, find the value of $\displaystyle e$.

    I haven't seen Impulse element ($\displaystyle I = mv - mu$) in collision questions before. How would I tackle this?
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    Quote Originally Posted by Air View Post
    Thanks Isomorphism!

    Also, there is another follow-on question, which is:

    c) Given that the magnitude of the impulse of $\displaystyle P$ on $\displaystyle Q$ is $\displaystyle \frac{32}{5} mu$, find the value of $\displaystyle e$.

    I haven't seen Impulse element ($\displaystyle I = mv - mu$) in collision questions before. How would I tackle this?
    Look at the change in velocity(while keeping sign in mind) of Q and hence momentum.this will give you I.

    Here the change in momentum is $\displaystyle mu + \frac15 mu (9e +4) = I = \frac{32}{5} mu$. Now solve for e.
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