A motionless 140 kg astronaut is holding a 15 kg tool while on a spacewalk. To get moving, the astronaut throws the tool forward at a speed of +4.2 m/s. How fast does the astronaut move backward?
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum of the system before throwing the tool should be equal to the total momentum after throwing the tool.
The total momentum before throwing the tool is given by the equation:
total momentum before = (mass of the astronaut) * (velocity of the astronaut) + (mass of the tool) * (velocity of the tool)
The total momentum after throwing the tool is given by the equation:
total momentum after = (mass of the astronaut) * (velocity of the astronaut after throwing the tool) + (mass of the tool) * (velocity of the tool after throwing the tool)
Since the astronaut is initially motionless, the velocity of the astronaut before throwing the tool is 0 m/s. The velocity of the tool before throwing it is +4.2 m/s. Therefore, the total momentum before throwing the tool simplifies to:
total momentum before = 0 + (mass of the tool) * (velocity of the tool)
Now, the total momentum after throwing the tool is given by:
total momentum after = (mass of the astronaut) * (velocity of the astronaut after throwing the tool) + (mass of the tool) * (velocity of the tool after throwing the tool)
Since the astronaut and the tool are separate objects after the throw, their velocities can be considered as opposite in direction. Therefore, the velocity of the tool after throwing it is -4.2 m/s. The velocity of the astronaut after throwing the tool can be represented as v m/s (backward).
Plugging these values into the equation for total momentum after throwing the tool, we have:
0 + (mass of the tool) * (-4.2) = (mass of the astronaut) * (v) + (mass of the tool) * (-4.2)
Simplifying the equation, we get:
0 = (mass of the astronaut) * (v)
To solve for v, we divide both sides of the equation by the mass of the astronaut:
0 = v
Therefore, the astronaut moves backward at a speed of 0 m/s.