Physics

Question

Rodney is trying out one of Santa's new games that consists of three pieces that blow apart if the wrong key is inserted into the door. The elf picks the wrong key and the board explodes. One piece of mass 2 kg goes off at an angle of 60° from the horizon at 10 m/s. A second piece of mass 1.2 kg travels at an angle of 180° and is moving at 15 m/s.

a) find the momentum of the third piece.

b) find the velocity of the third piece if it's mass is 3 kg.​

1 Answer

  • a) The momentum of the third piece is 19.1 kg m/s

    b) The velocity is 6.4 m/s at an angle of [tex]-65^{\circ}[/tex]

    Explanation:

    a)

    We can solve this problem by applying the law of conservation of momentum in 2D. In fact, the total momentum of the board before the explosion must be equal to the total momentum of the 3 pieces after the explosion.

    Before the explosion, the total momentum is zero, since the board is at rest. Resolving into the x- and y- components, this means that the two components of the momentum are zero:

    [tex]p_x = 0\\p_y = 0[/tex]

    After the explosion, we have:

    [tex]p_x = (2 kg)(10 m/s)( cos 60^{\circ}) + (1.2 kg)(15 m/s)(cos 180^{\circ})+p_{3x}[/tex]

    where [tex]p_{3x}[/tex] is the x-component of the momemtum of the 3rd piece. By solving,

    [tex]p_x = -8+p_{3x}[/tex]

    And since the x-momentum must be conserved,

    [tex]0=-8+p_{3x} \rightarrow p_{3x}=8 kg m/s[/tex]

    Similarly, along the y-axis,

    [tex]p_y = (2 kg)(10 m/s)( sin 60^{\circ}) + (1.2 kg)(15 m/s)(sin 180^{\circ})+p_{3y}[/tex]

    where [tex]p_{3y}[/tex] is the y-component of the momemtum of the 3rd piece. By solving,

    [tex]p_y = 17.3+p_{3y}[/tex]

    And since the y-momentum must be conserved,

    [tex]0=17.3+p_{3y} \rightarrow p_{3y}=-17.3 kg m/s[/tex]

    So now we can find the momentum of the third piece:

    [tex]p_3 = \sqrt{p_{3x}^2+p_{3y}^2}=\sqrt{8^2+(-17.3)^2}=19.1 kg m/s[/tex]

    b)

    The momentum of the third piece is given by

    [tex]p_3 = m v_3[/tex]

    where

    m = 3 kg is the mass of the piece

    [tex]v_3[/tex] is the velocity

    Solving for [tex]v_3[/tex], we find

    [tex]v_3 = \frac{p_3}{m}=\frac{19.1}{3}=6.4 m/s[/tex]

    We can also find the direction, which is identical to the direction of the momentum, by using the following equation:

    [tex]\theta = tan^{-1}(\frac{p_{3y}}{p_{3x}})=tan^{-1}(\frac{-17.3}{8})=-65^{\circ}[/tex]

    So, the third piece flies off at 6.4 m/s at an angle of [tex]-65^{\circ}[/tex].

    Learn more about momentum:

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