an 83.2 kg girl is standing on a frictionless ice surface through a 2.00 kg book at 12.6 m/s at what velocity does the girl move across the ice?

conservation of momentum

m1u1+m2u2=m1v1+m2v2

m1 = mass of first thing (girl)
m2 = mass of second thing (book)
u1 = initial velocity of first thing (girl)
u2 = initial velocity of second thing (book)
v1 = final velocity of first thing (girl)
v2 = final velocity of second thing (book)

if the book and girl both start at rest (0 m/s) see if you can find v1 now.

To find the velocity at which the girl moves across the ice, we can use the principle of conservation of momentum. The initial momentum of the system consisting of the girl and the book is equal to the final momentum of the system.

The initial momentum (before the girl throws the book) is given by:
Momentum_initial = Mass_girl × Velocity_girl + Mass_book × Velocity_book

Since the girl is initially at rest on the frictionless ice surface, her initial velocity (Velocity_girl) is zero. Therefore, the initial momentum simplifies to:
Momentum_initial = 0 + Mass_book × Velocity_book

The final momentum (after the girl throws the book) is given by:
Momentum_final = Mass_girl × Velocity_girl_final + Mass_book × Velocity_book_final

Since both the girl and the book move together after the book is thrown, they have the same final velocity (Velocity_book_final = Velocity_girl_final = V).

Using the principle of conservation of momentum, we can equate the initial and final momenta:
Momentum_initial = Momentum_final
0 + Mass_book × Velocity_book = Mass_girl × V + Mass_book × V

Now, let's plug in the given values:
Mass_girl = 83.2 kg
Mass_book = 2.00 kg
Velocity_book = 12.6 m/s

Substituting these values into the equation:
0 + 2.00 kg × 12.6 m/s = 83.2 kg × V + 2.00 kg × V

Simplifying the equation:
25.2 kg m/s = (83.2 kg + 2.00 kg) × V
25.2 kg m/s = 85.2 kg × V

Now, solve for V:
V = 25.2 kg m/s / 85.2 kg

Calculating the value:
V ≈ 0.296 m/s

Therefore, the girl moves across the ice at a velocity of approximately 0.296 m/s.