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a bomb sitting at rest on a table explodes into four pieces of equal mass. the first piece travels to the south at a velocity of 55.0 m/s. the second piece travels to the west at a velocity of 80.0 m/s. the third piece travels at a velocity of 40.0 m/s [30.0 w of n]. what is the velocity of the fourth piece?

Sagot :

onyebuchinnaji

The velocity of the fourth piece after the explosion is 118.5 m/s at 14.7⁰ east of north.

Conservation of linear momentum

The principle of conservation of linear momentum states that the sum of the initial momentum is equal to sum of final momentum.

Initial momentum of the bomb before explosion is equal to the final momentum of pieces of bomb after explosion.

Pi = Pf

0 = P₁ + P₂ + P₃ + P₄

0 = mv₁ + mv₂ + mv₃ + mv₄

0 = v₁ + v₂ + v₃ + v₄

Resolve the velocity into x and y components

Consider x-y plane for the velocity of the piece of bomb;

  • Let southward = negative direction
  • Let northward = positive direction
  • Let eastward = positive direction
  • Let westward = negative direction

[tex]0 = (v_1x + v_1y) + (v_2x + v_2_y) + (v_3x + v_3_y) + (v_4_x + v_4_y)\\\\0 = (0 - 50y) + (-80x + 0) + (-40cos30x + 40sin30y) + (v_4_x + v_4_y)\\\\0 = -50y -80x - 34.64x + 20y + v_4_x + v_4_y\\\\0 = -30y - 114.64x + v_4_x + v_4_y \\\\v_4_x + v_4_y = 114.64x + 30y\\\\|v_4| = \sqrt{114.64^2 + 30^2} \\\\|v_4| = 118.5 \ m/s\\\\tan(\theta ) = \frac{30}{114.64} \\\\tan(\theta) = 0.262\\\\\theta = tan^{-1} (0.262)\\\\\theta= 14.7 \ ^0[/tex]

Thus, the velocity of the fourth piece is 118.5 m/s at 14.7⁰ east of north.

Learn more about conservation of linear momentum here: https://brainly.com/question/7538238

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