An object with total mass m_total = 17.5 kg is sitting at rest when it explodes

Question

An object with total mass m_total = 17.5 kg is sitting at rest when it explodes into three pieces. One piece with mass m_1 = 4.5 kg moves up and to the left at an angle of theta_1 = 22 degree above the -x axis with a speed of v_1 = 26.4 m/s. A second piece with mass m_2 = 5.1 kg moves down and to the right an angle of theta_2 = 27 degree to the right of the -y axis at a speed of v_2 = 22.1 m/s. What is the magnitude of the final momentum of the system (all three pieces)? What is the mass of the third piece? What is the x-component of the velocity of the third piece? What is the y-component of the velocity of the third piece? What is the magnitude of the velocity of the center of mass of the pieces after the collision? Calculate the increase in kinetic energy of the pieces during the explosion.

Explanation / Answer

1) magnitude of final momentum will be zero because the inital momentum is zero

2) mass of the third piece = M- (m1+m2) = 17.5 - (4.5+5.1) = 7.9 kg

3) conserving the momentum in x- direction we have

4.5*(-26.4) cos 22 + 5.1 *22.1 * sin 27 + 7.9 Vx = 0

Vx = 7.465 m/s

4) cosnerving the momentum in y - direction we have

4.5*(26.4) sin 22 + 5.1 *-22.1 * cos 27 + 7.9 Vy = 0

Vy = 7.078 m/s

velocity of the third piece = sqrt(7.465^2 + 7.078^2) = 10.28 m/s

5) magnitude of velocity of COM

V = m1v1 + m2v2 + m3v3 / M

V = 4.5*26.4 + 5.1*22.1 + 7.9*10.28 / 17.5

V = 17.87 m/s

6) Increase in KE = 0.5 MV^2 = 0.5*17.5*17.87^2 = 2794.14 J

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