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
