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?
To answer this question, we can make use of the law of conservation of momentum. According to this law, the total momentum of two objects before and after an interaction remains constant, assuming no external forces act on them.
When the two astronauts push each other apart, they exert equal and opposite forces on each other. As a result, their momentum changes. Let's denote the mass of the lighter astronaut as m1 (61 kg) and the mass of the heavier astronaut as m2 (82 kg).
Initially, both astronauts are at rest, so their initial momentum is zero. After they push each other apart, the total momentum of the system is still zero because momentum is conserved. Therefore, the momentum gained by the lighter astronaut is equal in magnitude but opposite in direction to the momentum gained by the heavier astronaut.
We can use the following equation to express momentum:
m1 * v1 + m2 * v2 = 0
Where:
m1 and m2 are the masses of the astronauts,
v1 and v2 are the velocities of the lighter astronaut and the heavier astronaut, respectively.
Since we are given that the lighter astronaut moves 12 m, we need to find the distance the heavier astronaut moves, as they are pushed apart.
To solve this problem, we need to use the concept of impulse. Impulse is the change in momentum of an object and is equal to the force applied to an object multiplied by the time interval over which the force is applied. The equation for impulse is as follows:
Impulse = Force * Time
For the lighter astronaut, we can write the impulse as:
I1 = Force * Time
For the heavier astronaut, the impulse is equal in magnitude but opposite in direction:
I2 = -Force * Time
Since impulse is equal to the change in momentum, we can say:
I1 = m1 * Δv1
I2 = m2 * Δv2
Now, let's combine the equations for impulse and momentum to solve the problem.
m1 * Δv1 = Force * Time
m2 * Δv2 = -Force * Time
From here, we can eliminate the time interval (Time). Divide both equations, and we get:
(m1 * Δv1) / (m2 * Δv2) = -1
Now, let's substitute the given values. We have m1 = 61 kg, m2 = 82 kg, and Δv1 = 12 m/s.
(61 kg * 12 m/s) / (82 kg * Δv2) = -1
To find Δv2 (the velocity of the heavier astronaut), we rearrange the equation:
82 kg * Δv2 = (61 kg * 12 m/s) / -1
Now, solving for Δv2:
Δv2 = (61 kg * 12 m/s) / -82 kg
Calculating the equation:
Δv2 ≈ -8.976 m/s
The negative sign indicates that the heavier astronaut moves in the opposite direction of the lighter astronaut. Now, we have the velocities of both astronauts.
Next, we can calculate the distance (d) the heavier astronaut moves using the equation:
d = v * t
Since we are interested in the distance when the lighter astronaut has moved 12 m, we can find the time it takes for the heavier astronaut to cover that distance:
t = 12 m / Δv2
Substituting the known values:
t = 12 m / -8.976 m/s
Calculating the equation:
t ≈ -1.336 seconds
The negative sign is expected since the time represents the duration when the two astronauts are moving apart. It doesn't affect the final result of the distance, which is always positive.
Finally, we can calculate the distance (d) the heavier astronaut moves:
d = Δv2 * t
Substituting the known values:
d ≈ -8.976 m/s * -1.336 s
Calculating the equation:
d ≈ 12 m
So, when the lighter astronaut has moved 12 m, the heavier astronaut will also be 12 m away from their initial position.
The center of gravity has not moved.
(totalmass)0=82*d+61(12)
so d+12 will be how far they are apart.