Understanding the Concepts
* Conservation of Momentum: In a closed system (like an astronaut in space), the total momentum before an event equals the total momentum after the event. Momentum is calculated as mass times velocity (p = mv).
* Momentum Before: The astronaut is initially at rest, so their momentum is 0.
* Momentum After: The astronaut recoils in one direction, and the gas is ejected in the opposite direction.
Setting up the Equation
Let:
* `m1` = mass of the astronaut (50 kg)
* `m2` = mass of the gas (100 g = 0.1 kg)
* `v1` = recoil velocity of the astronaut (what we want to find)
* `v2` = velocity of the ejected gas (given, but not specified in the problem)
The conservation of momentum equation is:
`0 = m1 * v1 + m2 * v2`
Solving for the Recoil Velocity
1. Rearrange the equation:
`v1 = - (m2 * v2) / m1`
2. Plug in the values:
`v1 = - (0.1 kg * v2) / 50 kg`
3. Simplify:
`v1 = -0.002 * v2`
Important Note: You need to know the velocity (`v2`) at which the gas is ejected to calculate the astronaut's recoil velocity. The problem statement doesn't provide this value.
Example:
Let's say the gas is ejected at a velocity of 100 m/s. Then:
`v1 = -0.002 * 100 m/s = -0.2 m/s`
This means the astronaut would recoil in the opposite direction of the gas ejection with a velocity of 0.2 m/s.
