A space probe, initially at rest, undergoes an internalmechanical malfunction and breaks into three pieces. Onepiece of mass m 1 = 48.0 kg travels in the positive x - directionat 12.0 m/s, and a second piece of mass m 2 = 62.0 kg travelsin the xy-plane at an angle of 105° at 15.0 m/s. The third piecehas mass m 3 = 112 kg. (a) Sketch a diagram of the situation,labeling the different masses and their velocities. (b) Writethe general expression for conservation of momentum in thex- and y-directions in terms of m 1, m 2, m 3, v 1, v 2, and v 3 and thesines and cosines of the angles, taking θ to be the unknownangle. (c) Calculate the final x-components of the momentaof m 1 and m 2. (d) Calculate the final y-components of themomenta of m 1 and m 2. (e) Substitute the known momentumcomponents into the general equations of momentumfor the x- and y-directions, along with the known mass m 3.(f) Solve the two momentum equations for v 3 cos θ and v 3sin θ, respectively, and use the identity cos2 θ + sin2 θ = 1 toobtain v 3. (g) Divide the equation for v 3 sin u by that for v 3cosθ to obtain tan θ, then obtain the angle by taking the inversetangent of both sides. (h) In general, would three such piecesnecessarily have to move in the same plane? Why?
A space probe, initially at rest, undergoes an internal
mechanical malfunction and breaks into three pieces. One
piece of mass m 1 = 48.0 kg travels in the positive x - direction
at 12.0 m/s, and a second piece of mass m 2 = 62.0 kg travels
in the xy-plane at an angle of 105° at 15.0 m/s. The third piece
has mass m 3 = 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 m 1, m 2, m 3, v 1, v 2, and v 3 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 m 1 and m 2. (d) Calculate the final y-components of the
momenta of m 1 and m 2. (e) Substitute the known momentum
components into the general equations of momentum
for the x- and y-directions, along with the known mass m 3.
(f) Solve the two momentum equations for v 3 cos θ and v 3
sin θ, respectively, and use the identity cos2 θ + sin2 θ = 1 to
obtain v 3. (g) Divide the equation for v 3 sin u by that for v 3cosθ 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?
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