An object with total mass mtotal = 14 kg is sitting at rest when it explodes int

Question

An object with total mass mtotal = 14 kg is sitting at rest when it explodes into three pieces. One piece with mass m1 = 4.6 kg moves up and to the left at an angle of 1 = 20° above the x axis with a speed of v1 = 28.4 m/s. A second piece with mass m2 = 5.3 kg moves down and to the right an angle of 2 = 25° to the right of the -y axis at a speed of v2 = 20.5 m/s.

3)

What is the x-component of the velocity of the third piece?

4)

What is the y-component of the velocity of the third piece?  

5)

What is the magnitude of the velocity of the center of mass of the pieces after the collision?

6)

Calculate the increase in kinetic energy of the pieces during the explosion.

Explanation / Answer

we have initial momentum = 0
we have m3 = M - (m1 + m2) = 14 - (4.6 + 5.3) = 4.1 Kg
So, momentum of m1 = 4.6 * 28.4 * ( cos (20) i + sin(20) j) = 122.76144 i + 44.681511 j
momentum of m2 = 5.3 * 20.5 * ( sin(25) i - cos(25) j) = 45.917474 i - 98.470341 j
From conservation of momentum,
M*0 = momentum of m1 + momentum of m2 + momentum of m3
so, momentum of m3 = 0 - (momentum of m1 + momentum of m2) = 0 - (122.76144 i + 44.681511 j ) - (45.917474 i - 98.470341 j)
momentum of m3 = -168.678914 + 53.78883 j
so, velocity = momentum / mass = (-168.678914 + 53.78883 j )/4.1 = -41.141198 i + 13.1192268 j
3) x-component of the velocity of the third piece = 41.141198 m/s in the negative x- direction
4) y-component of the velocity of the third piece = 13.1192268 m/s in the positive y-direction
5) magnitude of the velocity of the center of mass of the pieces after the collision = 0
6) Initial kinetic energy = 0
Final kinetic energy of m1 = 0.5 * m1* v1^2 = 0.5 * 4.6 * (28.4^2) = 1855.088 J
Final kinetic energy of m2 = 0.5 * m2* v2^2 = 0.5 * 5.3 * (20.5^2) = 1113.6625 J
magnitude of velocity of m3 = sqrt ( 41.141198^2 + 13.1192268^2) = 43.182314 m/s
Final kinetic energy of m3 = 0.5 * m3*v3^2 = 0.5*4.1 * 43.182314^2 = 3822.66 J
Total final energy = 3822.66 + 1855.088 + 1113.6625 = 6791.4105 J is the increase in kinetic energy of the pieces during the explosion.

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