Two astronauts, one of mass 61 kg and the other 82 kg, are initially at rest in outer space. They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m?
Their center of mass will remain in the same place (in a coordinate system fixed with respect their initial location), since no external force acts upon them.
it will be smaller than smaller 12 m
To solve this problem, we will use the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant as long as no external forces act on it.
Step 1: Calculate the initial momentum of the system.
The initial momentum of the system is zero since both astronauts are initially at rest.
Step 2: Calculate the final momentum of the system.
The final momentum of the system can be calculated by multiplying the mass of each astronaut by their respective velocities. Since they are pushing each other apart, their velocities will have opposite signs.
Let's assume the velocity of the lighter astronaut (mass = 61 kg) after moving 12 m is v1, and the velocity of the heavier astronaut (mass = 82 kg) is v2.
The final momentum of the system is given by:
Final momentum = (mass of lighter astronaut * velocity of lighter astronaut) + (mass of heavier astronaut * velocity of heavier astronaut)
= (61 kg * v1) + (82 kg * v2)
Step 3: Apply the law of conservation of momentum.
According to the conservation of momentum, the initial momentum should be equal to the final momentum. Since the initial momentum was zero, we can set up the equation:
Initial momentum = Final momentum
0 = (61 kg * v1) + (82 kg * v2)
Step 4: Calculate the distance traveled by the heavier astronaut.
We are given that the lighter astronaut has moved 12 m, which means the heavier astronaut has moved some distance as well. Let's assume that distance as d.
Since distance traveled is equal to the velocity multiplied by the time taken, we have:
12 m = v1 * t1 ... (equation 1)
d = v2 * t2 ... (equation 2)
Step 5: Relate the velocities of both astronauts.
Since they push each other apart, the sum of their velocities after the push will be zero. Therefore, we can equate their velocities:
v1 + v2 = 0
Step 6: Substitute the values.
From equation 1, we can solve for t1:
t1 = 12 m / v1
From equation 2, we can solve for t2:
t2 = d / v2
Substituting these values back into the equation 0 = (61 kg * v1) + (82 kg * v2), we get:
0 = 61 kg * (12 m / v1) + 82 kg * (d / v2)
Simplifying the equation:
0 = 732 kg*m/v1 + 82 kg*d/v2
0 = 732/v1 + d/v2
Step 7: Solve for d (distance traveled by the heavier astronaut).
Now, we have to solve for d. To do this, we will use the relationship v1 + v2 = 0 and substitute v2 = -v1 into the equation:
0 = 732/v1 + d/(-v1)
0 = 732 - d/v1
Rearranging the equation:
d/v1 = 732
d = 732 * v1
Step 8: Substitute the value of d into equation 2 to solve for v2.
Using the value of d from step 7, equation 2 becomes:
732 * v1 = -v2 * t2
Substituting the value of d/v1 from step 7:
732 = -v2 * t2
Step 9: Substitute the value of t2 from equation 2 into equation 1.
Using the value of t2 from step 4, equation 1 becomes:
12 = v1 * (d/v2)
Substituting the value of d from step 7, and rearranging the equation:
12 = v1 * (732 / -v1)
Simplifying the equation:
12 = -732
Since this equation does not hold true, it means that the assumption made in step 7 is not correct.
Therefore, in this scenario, it is not possible to determine the distance traveled by the heavier astronaut when the lighter astronaut has moved 12 m, as it violates the principle of conservation of momentum.
To find out how far apart they are when the lighter astronaut has moved 12 m, we can use the law of conservation of momentum.
According to the law of conservation of momentum, the total momentum before the push is equal to the total momentum after the push.
Before the push, both astronauts are at rest, so their initial momentum is zero.
After the push, the total momentum should still be zero because there are no external forces acting on the astronauts.
Let's denote the distance the astronauts move apart as 'x'. When the lighter astronaut moves 12 m to the right, the heavier astronaut moves 'x' meters to the left.
Using the law of conservation of momentum, we can set up an equation:
(mass of the lighter astronaut) * (velocity of the lighter astronaut) + (mass of the heavier astronaut) * (velocity of the heavier astronaut) = 0
Since they are initially at rest, both velocities are zero, so the equation becomes:
(61 kg) * (0 m/s) + (82 kg) * (0 m/s) = 0
Simplifying the equation, we get:
0 + 0 = 0
Therefore, we have no useful information from the conservation of momentum equation.
To find out how far apart they are when the lighter astronaut has moved 12 m, we need to consider the concept of relative motion. Since the lighter astronaut has moved 12 m to the right and the heavier astronaut has moved to the left by the same distance, the total distance they are apart is the sum of these two distances.
Distance apart = Distance moved by the lighter astronaut + Distance moved by the heavier astronaut
Distance apart = 12 m + (-12 m)
Distance apart = 0 m
Therefore, when the lighter astronaut has moved 12 m, they are still at the same position, 0 meters apart.