A 58-kg astronaut inside a 9-m-long space capsule of mass 550 kg is floating weightlessly on one end of the capsule. He kicks off the wall at a velocity of 3.6 m/s toward the other end of the capsule. How long does it take the astronaut to reach the far wall?
To find the time it takes for the astronaut to reach the far wall, we can use the principle of conservation of linear momentum.
The initial linear momentum of the astronaut and the capsule system is zero, considering they are floating weightlessly. When the astronaut kicks off the wall, he gains linear momentum in the opposite direction. According to the law of conservation of linear momentum, the total linear momentum before and after the kick must remain constant.
The linear momentum of an object can be calculated using the formula:
Linear Momentum = mass × velocity
Given that the astronaut has a mass of 58 kg and a velocity of 3.6 m/s, the initial linear momentum after the kick is:
Initial Linear Momentum = 58 kg × (-3.6 m/s) since the astronaut is moving in the opposite direction
The capsule has a mass of 550 kg and is initially at rest, so its initial linear momentum is zero.
The total initial linear momentum of the system (astronaut + capsule) is therefore:
Total Initial Linear Momentum = Initial Linear Momentum of Astronaut + Initial Linear Momentum of Capsule
= 58 kg × (-3.6 m/s) + 0
= -208.8 kg·m/s
According to the conservation of linear momentum, this total initial linear momentum must be equal to the total final linear momentum of the system.
Since the astronaut moves to the other end of the capsule, the capsule will move in the opposite direction with some final velocity. Let's denote this final velocity as Vf.
The final linear momentum can be calculated as:
Final Linear Momentum = (Mass of Astronaut + Mass of Capsule) × Final Velocity
Given that the mass of the astronaut is 58 kg, the mass of the capsule is 550 kg, and the final velocity of the capsule is Vf (which is the velocity of the astronaut with the opposite sign), the final linear momentum is:
Final Linear Momentum = (58 kg + 550 kg) × (-Vf)
= 608 kg × (-Vf)
= -608Vf kg·m/s
Since the initial and final linear momenta are equal due to the conservation of linear momentum, we have:
Total Initial Linear Momentum = Total Final Linear Momentum
-208.8 kg·m/s = -608Vf kg·m/s
Dividing both sides of the equation by -608 kg·m/s, we get:
Vf = -208.8 kg·m/s / -608 kg·m/s
≈ 0.3434 m/s
Therefore, the final velocity of the capsule is approximately 0.3434 m/s.
Now, to find the time it takes for the astronaut to reach the far wall, we can use the following equation of motion:
Final Velocity = Initial Velocity + (Acceleration × Time)
Since the astronaut's initial velocity is 3.6 m/s, the final velocity is -0.3434 m/s (opposite direction), and the acceleration can be calculated using Newton's second law:
Force = mass × acceleration
The force exerted on the astronaut can be determined by considering the reaction force from the wall. When the astronaut kicks off the wall, the wall exerts an equal and opposite force on the astronaut, causing acceleration in the capsule-astronaut system.
The force exerted by the wall can be calculated using the formula:
Force = mass × acceleration
Given that the astronaut has a mass of 58 kg, the acceleration is computed as:
Force = 58 kg × acceleration
The mass of the capsule is 550 kg, and the total force acting on the system is due to the astronaut's kick, so the force on the wall is equal to the force exerted by the astronaut:
Force = Force on Wall = 58 kg × acceleration
Furthermore, Newton's third law states that for every action, there is an equal and opposite reaction. Hence, the force exerted by the wall (Force on Wall) is the same as the force experienced by the astronaut (Force on Astronaut):
Force on Astronaut = Force on Wall = 58 kg × acceleration
Now, we know that acceleration = Force / mass, so we can express the acceleration in terms of the mass and force:
acceleration = (Force on Astronaut) / mass
= (58 kg × acceleration) / 58 kg
= acceleration
Substituting this acceleration in the equation of motion, we get:
Final Velocity = Initial Velocity + (acceleration × Time)
Rearranging the equation to solve for Time:
Time = (Final Velocity - Initial Velocity) / acceleration
Substituting the known values:
Time = (-0.3434 m/s - 3.6 m/s) / acceleration
= -3.9434 m/s / acceleration
Now, we can substitute the correct value of acceleration:
Time = -3.9434 m/s / acceleration
Since acceleration = (Force on Astronaut) / mass, we can substitute:
Time = -3.9434 m/s / ((58 kg × acceleration) / 58 kg)
= -3.9434 m/s / (Force on Astronaut / 58 kg)
So, the time it takes for the astronaut to reach the far wall is:
Time = -3.9434 m/s / (Force on Astronaut / 58 kg)