A space probe, initially at rest, undergoes an internal mechanical malfunction and breaks into three pieces. One piece of mass ml = 48.0 kg travels in the positive x-direction at 12.0 m/s, and a second piece of mass m2 = 62.0 kg travels in the xy-plane at an angle of 105° at 15.0 m/s. The third piece has mass m3 = 112 kg. (a) Sketch a diagram of the situation, labeling the different masses and their velocities, (b) Write the general expression for conservation of momentum in the x- and y-directions in terms of m1, m2, m3, v1, v2 and v3 and the sines and cosines of the angles, taking θ to be the unknown angle, (c) Calculate the final x-components of the momenta of m1 and m2. (d) Calculate the final y-components of the momenta of m1 and m2. (e) Substitute the known momentum components into the general equations of momentum for the x- and y-directions, along with the known mass m3. (f) Solve the two momentum equations for v3 cos θ and v3 sin θ, respectively, and use the identity cos2 θ + sin2 θ = 1 to obtain v3. (g) Divide the equation for v3 sin θ by that for v3 cos θ to obtain tan θ, then obtain the angle by taking the inverse tangent of both sides, (h) In general, would three such pieces necessarily have to move in the same plane? Why?
a)
Answer to Problem 46P
Explanation of Solution
The diagram of the breakage is,
The numerical values of the masses and velocities are,
The numerical values of the masses and velocities are,
Conclusion:
Thus, the diagram of the breakage is,
(b)
Answer to Problem 46P
Explanation of Solution
The general expression for the conservation of momentum in x-direction is,
The general expression for the conservation of momentum in y-direction is,
Conclusion:
Thus, the general expression for the conservation of momentum in x-direction is
(c)
Answer to Problem 46P
Explanation of Solution
The final x-component of the momenta of the mass
Substitute
The final x-component of the momenta of the mass
Substitute
Conclusion:
Thus, the final x-component of the momenta of the mass
(d)
Answer to Problem 46P
Explanation of Solution
The final y-component of the momenta of the mass
Substitute
The final y-component of the momenta of the mass
Substitute
Conclusion:
Thus, the final y-component of the momenta of the mass
(e)
Answer to Problem 46P
Explanation of Solution
The final x-component of the momenta of mass
The final y-component of the momenta of mass
Conclusion:
Thus, the final x and y-components of the momenta of mass
(f)
Answer to Problem 46P
Explanation of Solution
In the x-direction,
In the y-direction,
Squaring and adding the equations,
Conclusion:
Thus, the velocity
(g)
Answer to Problem 46P
Explanation of Solution
The tangent of the angle is,
Thus, the angle is,
The angle must be in third quadrant. So, angle
Conclusion:
Thus, the angle
(h)
Answer to Problem 46P
Explanation of Solution
The momentum of the third fragment must be equal in magnitude and must be in the opposite direction to the resultant of the other two fragments momenta. So, all three pieces have to move in the same plane.
Conclusion:
Thus, all three pieces have to move in the same plane.
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