A spherical satellite of approximately uniform density with radius 6.9 m and mas

Question

A spherical satellite of approximately uniform density with radius 6.9 m and mass 260 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 9.6 kg is initially moving toward the satellite with velocity vector (-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 vector (-1300,480,0) m/s. Both before and after the collision, the rotation of the space junk is negligible.

(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.)

(b) Calculate the rise in the internal energy of the satellite and space junk combined.
J

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 )

(260)(2600 i + 0 j ) + 9.6 (- 2200 i + 0 j) = (260) (Vx i + Vy j) + (9.6) (- 1300 i + 480 j )

676000 i - 21120 i = (260) Vx i + (260) Vy j - 12480 i + 4608 j

Vx i + Vy j = 2648 i - 98.95 j

Vx = 2648 m/s

Vy = - 98.95 m/s

I = moment of inertia = (0.4) MR2 = (0.4) (260) (6.9)2 = 4951.44 kgm2

using conservation of angular momentum

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

- (9.6) (2200) (6.9) + (4951.44) (2) = - (9.6) (6.9) (2649.85) + (4951.44) w'

w' = 8.02 rad/s

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