An 84.9 kg man is standing on a frictionless ice surface when he throws a 2.00kg book at 11.1m/s. With what velocity does the man move across the ice?
final momentum = initial momentum
book moves in + direction, man moves in - direction
2*11.1 + 84.9 v = 0
v = -22.1/84.9
To find the velocity with 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.
The momentum of an object is calculated by multiplying its mass by its velocity. Therefore, the momentum of the man before the book is thrown can be calculated as:
Momentum of man before = mass of man × velocity of man before book is thrown
Given:
Mass of the man (m1) = 84.9 kg
Velocity of the man before the book is thrown (v1) = ?
Now, let's denote the momentum of the book as (momentum of book = m2v2), where m2 is the mass of the book and v2 is its velocity after it is thrown.
We know that the total momentum before the book is thrown should be equal to the total momentum after the book is thrown. Mathematically, it can be represented as:
Momentum of man before = Momentum of man after + Momentum of book
Using the equation of momentum (mass × velocity), we can rewrite the equation as:
m1v1 = (m1 + m2) × v3
Here, v3 is the velocity with which the man moves across the ice after throwing the book.
Solving for v3, we get:
v3 = (m1v1) / (m1 + m2)
Now we can substitute the given values to find the velocity of the man:
v3 = (84.9 kg × 0 m/s) / (84.9 kg + 2.00 kg)
Since the man is standing still (initially at rest), the velocity of the man before throwing the book can be considered as 0 m/s.
v3 = 0 m/s
Therefore, the velocity with which the man moves across the ice after throwing the book is 0 m/s or, simply put, the man remains at rest on the ice.
To solve this problem, we can use the law of conservation of momentum which states that the total momentum before an event is equal to the total momentum after the event.
The momentum of an object is given by the equation:
momentum = mass × velocity
Before the man throws the book, the total momentum is zero because both the man and the book are at rest.
After the man throws the book, the total momentum should still be zero because there are no external forces acting on the system.
Let's assume the velocity at which the man moves after throwing the book is V.
The momentum of the man after throwing the book is given by:
momentum_man = mass_man × velocity_man
= 84.9 kg × V
The momentum of the book after being thrown is given by:
momentum_book = mass_book × velocity_book
= 2.00 kg × 11.1 m/s
According to the law of conservation of momentum, the total momentum before and after the event should be the same. Therefore, we can set up the following equation:
momentum_man + momentum_book = 0
Substituting the known values into the equation:
84.9 kg × V + 2.00 kg × 11.1 m/s = 0
Solving for V:
84.9 kg × V = -2.00 kg × 11.1 m/s
V = -2.00 kg × 11.1 m/s / 84.9 kg
Calculating the value of V:
V ≈ -0.260 m/s (rounded to the nearest thousandth)
Therefore, the man moves across the ice with a velocity of approximately -0.260 m/s in the opposite direction of the book's velocity.