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.

In the of conservation of momentum if 2 objects collide and recoils or bounces from each other, they must have the same amount of momentum.

The momentum of the oxygen tank is 3.0kg x 15 m/s = 45. The Austronoaut should have the same momentum by dividing her mass with the momentum of the tank which is 80/45 is .5625 m/s.
.5625 is the velocity of the austronet during or after her recoil and if you try to get her momentum by multiplying her mass and velocity, it would turn out to be 45 units of mementum, becuse the system made should have the same momentum for each object colliding and using the law to conservation of momentum conserve it or equalize it.

1. 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 no external forces are acting.

The momentum of an object is defined as the product of its mass and velocity, written as p = mv. In this case, the momentum of the astronaut prior to throwing the oxygen tank is given by:

p1 = m1 * v1

Where m1 is the mass of the astronaut and v1 is her initial velocity (which is zero since she is at rest).

The momentum of the oxygen tank after it is thrown is given by:

p2 = m2 * v2

Where m2 is the mass of the oxygen tank (3.0 kg) and v2 is its velocity (given as 15 m/s in the opposite direction of the spaceship).

Since the total momentum before the event is zero (the astronaut and oxygen tank are at rest), the total momentum after the event should also be zero. Therefore, we can set up the following equation:

0 = p1 + p2

0 = m1 * v1 + m2 * v2

Substituting the given values:

0 = 80.0 kg * 0 + 3.0 kg * (-15 m/s)

Simplifying:

0 = 0 - 45 kg*m/s

Now, we can solve for the velocity of the astronaut (v1):

45 kg*m/s = 80.0 kg * v1

v1 = 45 kg*m/s / 80.0 kg

v1 = 0.5625 m/s

Therefore, the astronaut will recoil towards the spaceship with a velocity of 0.5625 m/s.

2. To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting.

The momentum of an object is defined as the product of its mass and velocity, written as p = mv. In this case, the momentum of the arrow before the collision is given by:

p1 = m1 * v1

Where m1 is the mass of the arrow (0.08 kg) and v1 is its initial velocity (given as 80.0 m/s).

The momentum of the block and the arrow after the collision is given by:

p2 = m2 * v2

Where m2 is the mass of the block (10.0 kg) and v2 is their common velocity after the collision.

Since the total momentum before the collision should be equal to the total momentum after the collision, we can set up the following equation:

p1 = p2

m1 * v1 = m2 * v2

Substituting the given values:

(0.08 kg) * (80.0 m/s) = (10.0 kg + 0.08 kg) * v2

Simplifying:

6.4 kg*m/s = (10.08 kg) * v2

Now, we can solve for the velocity of the block and arrow (v2):

v2 = 6.4 kg*m/s / 10.08 kg

v2 ≈ 0.6349 m/s

Therefore, the speed of the block and arrow just after the collision is approximately 0.6349 m/s.

To answer these questions, we can use the conservation of momentum principle. According to this principle, the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.

1. For the first question:
To find the speed at which the astronaut recoils toward the spaceship, we can apply the conservation of momentum principle. Let's denote the initial momentum of the system (astronaut + tank) as P_initial and the final momentum as P_final.

The initial momentum (P_initial) is the momentum of the astronaut before she throws the oxygen tank. Since she is at rest, her momentum is zero. The oxygen tank has a mass of 3.0 kg and is moving away from the spaceship at a speed of 15 m/s. Therefore, the initial momentum is:
P_initial = 0 + (3.0 kg)(-15 m/s) = -45 kg·m/s (to the left, opposite direction to the motion of the tank)

After throwing the oxygen tank, the only object left in the system is the astronaut. Let's denote her final speed as v_final. Since there are no external forces acting on the system, the momentum before and after the throw must be equal:
P_initial = P_final

Using this principle, we can write down the equation:
-45 kg·m/s = (80.0 kg + 3.0 kg) * v_final

Solving this equation will give us the speed at which the astronaut recoils toward the spaceship.

2. For the second question:
To find the speed of the block and arrow just after the collision, we can again apply the conservation of momentum principle.

The initial momentum of the system (arrow + block) is the momentum of the arrow before the collision. The mass of the arrow is 0.08 kg, and its initial speed is 80.0 m/s. Therefore, the initial momentum is:
P_initial = (0.08 kg)(80.0 m/s) = 6.4 kg·m/s

After the collision, the arrow embeds into the block, so the final momentum of the system (arrow + block) is the momentum of the combined mass (arrow + block). Let's denote the final speed of the block and arrow as v_final.

Using the principle of conservation of momentum, we can write the equation:
P_initial = (0.08 kg + 10.0 kg) * v_final

Solving this equation will give us the speed of the block and arrow just after the collision.

By using these principles and equations, we can find the answers to the given questions.