A spherical satellite of approximately uniform density with radius 6.3 m and mas

Question

A spherical satellite of approximately uniform density with radius 6.3 m and mass 270 kg is originally moving with velocity (2600, 0, 0 m/s, and is originally rotating with an angular speed 2 rad/s, in the direction shown in the diagram. A small piece of space junk of mass 9.1 kg is initially moving toward the satellite with velocity -2200, 0, 0> m/s. The space junk hits the edge of the satellite as shown in the figure below, and moves off with a new velocity (-1300, 480, 0) m/s. Both before and after the collision, the rotation of the space junk is negligible. V3 M, R,I (a) Just after the collision, what are the components of the center-of-mass velocity of the satellite (Vx and Vy) and its rotational speed (For vx, enter your answer to at least four significant figures.) Vy= m/s m/s rad/s b) Calculate the rise in the internal energy of the satellite and space junk combined

Explanation / Answer

a)

using conservation of linear momentum

M(2600 i + 0 j ) + m (- 2200 i + 0 j) = M (Vx i + Vy j) + m (- 1300 i + 480 j )

(270)(2600 i + 0 j ) + 9.1 (- 2200 i + 0 j) = (270) (Vx i + Vy j) + (9.1) (- 1300 i + 480 j )

702000 i - 20020 i = (270) Vx i + (270) Vy j - 11830 i + 4368 j

Vx i + Vy j = 2643.8 i - 90.33 j

Vx = 2643.8 m/s

Vy = - 90.33 m/s

I = moment of inertia = (0.4) MR2 = (0.4) (270) (6.3)2 = 4286.52 kgm2

using conservation of angular momentum

- m v2 R + I w = - m v3 R + I w'

- (9.1) (2200) (6.3) + (4286.52) (2) = - (9.1) (6.3) (1385.8) + (4286.52) w'

w' = - 8.88 rad/s

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