An 84.9 kg man is standing on a frictionless ice surface when he throws a 2.00 kg book at 11.1 m/s. With what velocity does the man move across the ice?

To determine the velocity at which the man moves across the ice, we can use the principle of conservation of momentum. According to this principle, the total momentum before the book is thrown should be equal to the total momentum after the book is thrown.

Let's assume the initial velocity of the man is u m/s, and the final velocity of the man is v m/s. The momentum of an object is given by the product of its mass and velocity. Therefore, the initial momentum of the man is the product of his mass (84.9 kg) and his initial velocity (u), which is equal to 84.9u kg⋅m/s.

After the man throws the book, the momentum of the man and the momentum of the book must add up to a total momentum of zero since there is no external force acting on the system to change its momentum. The momentum of the man is now the product of his mass (84.9 kg) and his final velocity (v), which is equal to 84.9v kg⋅m/s. The momentum of the book is the product of its mass (2.00 kg) and its velocity (11.1 m/s), which is equal to 22.2 kg⋅m/s.

According to the principle of conservation of momentum, the sum of the initial momenta should be equal to the sum of the final momenta:

Initial momentum = Final momentum
84.9u = 84.9v + 22.2

Now we can solve this equation for the final velocity (v):

84.9u - 22.2 = 84.9v
84.9v = 84.9u - 22.2
v = (84.9u - 22.2) / 84.9

Therefore, the velocity at which the man moves across the ice (v) can be calculated using the above equation.

It's important to note that an additional equation is required to solve for the final velocity. It could be an equation that describes a relationship between the initial and final velocities, such as the kinematic equation that relates the displacement, time, and acceleration of an object. However, without additional information or equations, it is not possible to determine the exact value of the final velocity.

To answer this question, we can apply the principle of conservation of linear momentum.

The momentum of an object is the product of its mass and velocity. The total momentum of a system remains constant if no external forces act on it. In this case, the man and the book make up the system.

Given:
- Mass of the man (m1) = 84.9 kg
- Mass of the book (m2) = 2.00 kg
- Velocity of the book (v2) = 11.1 m/s

Let's assume that the initial velocity of the man (v1i) is zero since he is standing still. The final velocity of the man (v1f) is what we need to find.

According to the principle of conservation of linear momentum:

Initial momentum = Final momentum

(mass of the man x initial velocity of the man) + (mass of the book x initial velocity of the book)
= (mass of the man x final velocity of the man) + (mass of the book x final velocity of the book)

(84.9 kg x 0 m/s) + (2.00 kg x 0 m/s) = (84.9 kg x final velocity of the man) + (2.00 kg x 11.1 m/s)

0 + 0 = 84.9 kg x final velocity of the man + 22.2 kg·m/s

Rearranging the equation:

84.9 kg x final velocity of the man = -22.2 kg·m/s
final velocity of the man = (-22.2 kg·m/s) / 84.9 kg

Calculating the final velocity of the man:

final velocity of the man = -0.2617 m/s

Hence, the man moves across the ice with a velocity of approximately -0.2617 m/s. The negative sign indicates that the man moves in the opposite direction of the book's velocity.