a 62.8 kg astronaut is floating toawrd the front of her ship at .16 m/s relative to the ship. she wishes to throw away the 2.9 kg book she was carrying. what should the speed of the book be to achive her goal?

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.

Let's denote the astronaut's mass as MA = 62.8 kg, the book's mass as MB = 2.9 kg, the initial velocity of the astronaut relative to the ship as VA = 0.16 m/s, and the final velocity of the astronaut and the book after she throws it as VF.

The initial momentum of the system is the sum of the momentum of the astronaut and the book:
Initial momentum = (MA + MB) * VA

The final momentum of the system is the sum of the momentum of the astronaut and the momentum of the book:
Final momentum = MA * VF + MB * VB

Since the sum of the initial momentum and the final momentum should be equal, we can set up the following equation:

(MA + MB) * VA = MA * VF + MB * VB

Substituting the given values, we have:

(62.8 kg + 2.9 kg) * 0.16 m/s = 62.8 kg * VF + 2.9 kg * VB

Simplifying the equation:

(65.7 kg) * 0.16 m/s = 62.8 kg * VF + 2.9 kg * VB

Now, the astronaut wishes to throw away the book, which means the final velocity of the book (VB) will be zero. Therefore, the equation becomes:

(65.7 kg) * 0.16 m/s = 62.8 kg * VF + 0

Rearranging the equation to solve for VF:

(65.7 kg) * 0.16 m/s - 0 = 62.8 kg * VF

10.512 kg m/s = 62.8 kg * VF

Dividing both sides of the equation by 62.8 kg:

(10.512 kg m/s) / 62.8 kg = VF

VF = 0.167 m/s

Therefore, to achieve her goal, the book should have a final speed of approximately 0.167 m/s.