An astronaut in her space suit has a total mass of m1 = 69.7 kg, including suit and oxygen tank. Her tether line loses its attachment to her spacecraft while she's on a spacewalk. Initially at rest with respect to her spacecraft, she throws her oxygen tank of mass m2 = 12.0-kg away from her spacecraft with a speed v = 7.60 m/s to propel herself back toward it

Determine the maximum distance she can be from the craft and still return within 2.40 min (the amount of time the air in her helmet remains breathable)
Explain in terms of Newton's laws of motion why this strategy works.

To determine the maximum distance the astronaut can be from the spacecraft and still return within 2.40 minutes, we can use the concept of conservation of momentum.

According to Newton's third law of motion, for every action, there is an equal and opposite reaction. When the astronaut throws her oxygen tank, she exerts a force on it, causing it to move in one direction. As a result, according to Newton's third law, the tank exerts an equal and opposite force on the astronaut, causing her to move in the opposite direction.

The momentum of an object is defined as the product of its mass and velocity. In this scenario, both the astronaut and the oxygen tank have momentum before the throw. However, since momentum is conserved, the total momentum after the throw should be equal to the total momentum before the throw.

Mathematically, we can write the conservation of momentum equation as:

m1v1 + m2v2 = (m1 + m2)vf

Where
m1 = mass of the astronaut (69.7 kg)
v1 = initial velocity of the astronaut (0 m/s - at rest)
m2 = mass of the oxygen tank (12.0 kg)
v2 = velocity of the oxygen tank (7.60 m/s - thrown by the astronaut)
vf = final velocity of the astronaut (unknown)

Using this equation, we can solve for the final velocity (vf) of the astronaut. Once we have the final velocity, we can calculate the maximum distance the astronaut can be from the spacecraft within 2.40 minutes.

Now, to calculate the maximum distance, we need to consider the time limit of 2.40 minutes (or 144 seconds). Since the astronaut is thrown in one direction and needs to return to the spacecraft, she will have a net displacement of zero. This means that the total time spent in moving away from the spacecraft should be equal to the total time spent in moving back towards it.

Let's assume that the astronaut spends t1 time moving away from the spacecraft and t2 time moving back towards it. Therefore:

t1 + t2 = 144 seconds

Now, we can set up two equations to calculate the maximum distance:

d1 = v1 * t1 (distance moved away from the spacecraft)
d2 = vf * t2 (distance moved back towards the spacecraft)

Where
v1 = initial velocity of the astronaut (0 m/s)
vf = final velocity of the astronaut (calculated using conservation of momentum equation)
t1 = time spent moving away from the spacecraft
t2 = time spent moving back towards the spacecraft
d1 = distance moved away from the spacecraft
d2 = distance moved back towards the spacecraft

Since the net displacement is zero, d1 = d2. Therefore:

v1 * t1 = vf * t2

Using this equation and the equation t1 + t2 = 144 seconds, we can solve for the maximum distance the astronaut can be from the spacecraft.

In terms of Newton's laws of motion, this strategy of throwing the oxygen tank works because of the conservation of momentum. By exerting a force on the tank and throwing it in one direction, the astronaut experiences an equal and opposite force, causing her to move in the opposite direction. This allows her to propel herself back towards the spacecraft and return to it within the time limit.