An astronaut is performing a space walk outside the International Space Station. The total mass of astronaut with her space suit and all lher gear is 127 kg. A small leak develps in her pulsion system and 5.84 g of gas are ejected each second into space with a speed of 754 m/s. She notices the leak 8.61 s after he start. How much will the gas leak have caused the from her original location in space by that time?
Assume astronaut starts form rest and the acceleration is constant...
To determine how much the gas leak will have caused the astronaut to move from her original location in space, we can use the principles of Newtonian physics.
First, let's calculate the total momentum change caused by the gas leak. The change in momentum can be found using the formula:
Δp = m * Δv
where Δp is the change in momentum, m is the mass of the ejected gas, and Δv is the change in velocity.
The mass of the ejected gas is given as 5.84 g or 0.00584 kg.
The change in velocity can be calculated by multiplying the speed of the ejected gas by the time it is flowing:
Δv = v * t
where v is the speed of the ejected gas and t is the time it is flowing.
The speed of the ejected gas is given as 754 m/s, and the time it is flowing is given as 8.61 s.
Now, let's calculate the change in momentum:
Δp = 0.00584 kg * (754 m/s * 8.61 s)
Calculating this, we find that the change in momentum is approximately 37.87 kg·m/s.
According to Newton's third law of motion, for every action, there is an equal and opposite reaction. Therefore, the total change in momentum experienced by the astronaut and her equipment must be equal and opposite to the change in momentum of the ejected gas:
Δp_tot = -Δp_gas
To find the change in velocity of the astronaut and her equipment, we can use the equation:
Δv_tot = Δp_tot / (m_astronaut + m_gear)
where Δv_tot is the change in velocity of the astronaut and her equipment, m_astronaut is the mass of the astronaut, and m_gear is the combined mass of her spacesuit and gear.
The mass of the astronaut is given as 127 kg, and we'll assume the mass of her spacesuit and gear remains constant.
Now, let's calculate the change in velocity of the astronaut and her equipment:
Δv_tot = -((-37.87 kg·m/s) / (127 kg))
Simplifying this expression, we find that the change in velocity of the astronaut and her equipment is approximately 0.298 m/s.
To determine how much the gas leak has caused the astronaut to move from her original location, we need to calculate the displacement. The displacement can be found using the equation:
Δx = v_0 * t + (1/2) * a * t^2
where Δx is the displacement, v_0 is the initial velocity (which is zero in this case), t is the time, and a is the acceleration.
We are given that the astronaut starts from rest, so the initial velocity is zero. We also know the time the astronaut notices the leak is 8.61 s.
To find the acceleration, we can use the equation:
a = Δv_tot / t
Substituting the known values:
a = 0.298 m/s / 8.61 s
Calculating this, we find that the acceleration is approximately 0.0346 m/s^2.
Now, let's calculate the displacement:
Δx = 0 * 8.61 s + (1/2) * 0.0346 m/s^2 * (8.61 s)^2
Simplifying this expression, we find that the displacement is approximately 1.371 meters.
Therefore, the gas leak will have caused the astronaut to move approximately 1.371 meters from her original location in space by the time she notices the leak.