1. An 80.0-kg astronaut is floating at rest a distance of 10.0 m from a spaceship when she runs out of oxygen and fuel to power her back to the spaceship. She removes her oxygen tank (3.0 kg) and flings it into space away from the ship with a speed of 15 m/s. At what speed does she recoil toward the spaceship?
2. An 0.08-kg arrow moving at 80.0 m/s hits and embeds in a 10.0-kg block resting on ice. Use the conservation-of-momentum principle to determine the speed of the block and arrow just after the collision.
To find the answers to these questions, we can use the concept of conservation of momentum. According to the principle of conservation of momentum, the total momentum before a collision is equal to the total momentum after the collision, assuming that no external forces are acting on the system.
1. Let's start with the first question:
An 80.0 kg astronaut is initially at rest a distance of 10.0 m from the spaceship. She removes her oxygen tank (3.0 kg) and flings it into space with a speed of 15 m/s. We need to find the speed at which she recoils toward the spaceship.
To solve this problem, we can consider the initial momentum of the astronaut-spaceship system before she flings the oxygen tank and the final momentum after the oxygen tank is flung.
The initial momentum of the system is given as:
Initial momentum = (Mass of astronaut) * (Velocity of astronaut)
Since the astronaut is initially at rest, her velocity is zero. Hence, the initial momentum will be zero.
The final momentum of the system is given as:
Final momentum = (Mass of astronaut) * (Velocity of astronaut) + (Mass of oxygen tank) * (Velocity of oxygen tank)
We know the mass and velocity of the astronaut, but we need to determine the velocity of the oxygen tank using the given information.
The conservation of momentum equation can be written as:
Initial momentum = Final momentum
0 = (80.0 kg) * (Velocity of astronaut) + (3.0 kg) * (Velocity of oxygen tank)
We have two unknowns, the velocity of the astronaut and the velocity of the oxygen tank. But if we observe carefully, we can see that the oxygen tank is flung into space and moves away from the spaceship. Hence, its velocity will be positive. Therefore, we can rewrite the equation as:
0 = (80.0 kg) * (Velocity of astronaut) + (3.0 kg) * (15 m/s)
Now we can solve for the velocity of the astronaut:
(80.0 kg) * (Velocity of astronaut) = -(3.0 kg) * (15 m/s)
Simplifying:
(80.0 kg) * (Velocity of astronaut) = -45 kg m/s
Dividing both sides by 80.0 kg:
Velocity of astronaut = -45 kg m/s / 80.0 kg
Calculating the value:
Velocity of astronaut = -0.563 m/s (rounded to three decimal places)
Therefore, the astronaut will recoil toward the spaceship with a speed of approximately 0.563 m/s.
2. Moving on to the second question:
An 0.08 kg arrow is moving at 80.0 m/s and hits and embeds in a 10.0 kg block resting on ice. We need to determine the speed of the block and arrow just after the collision.
Again, we'll use the conservation of momentum principle to solve this problem. The key idea is that the momentum before the collision is equal to the momentum after the collision.
The initial momentum of the system is given as:
Initial momentum = (Mass of arrow) * (Velocity of arrow)
The final momentum of the system is given as:
Final momentum = (Mass of arrow + Mass of block) * (Velocity of final system)
Using the conservation of momentum equation, we can equate the initial and final momenta:
(Mass of arrow) * (Velocity of arrow) = (Mass of arrow + Mass of block) * (Velocity of final system)
Substituting the given values:
(0.08 kg) * (80.0 m/s) = (0.08 kg + 10.0 kg) * (Velocity of final system)
Solving for the velocity of the final system:
(0.08 kg) * (80.0 m/s) = (10.08 kg) * (Velocity of final system)
Simplifying:
6.4 kg m/s = 10.08 kg * (Velocity of final system)
Dividing both sides by 10.08 kg:
Velocity of final system = 6.4 kg m/s / 10.08 kg
Calculating the value:
Velocity of final system = 0.635 m/s (rounded to three decimal places)
Hence, the speed of the block and arrow just after the collision is approximately 0.635 m/s.