A 83.8 kg astronaut is working on the en- gines of a spaceship that is drifting through space with a constant velocity. The astronaut turns away to look at Earth and several sec- onds later is 39.6 m behind the ship, at rest relative to the spaceship. The only way to re- turn to the ship without a thruster is to throw a wrench directly away from the ship. The wrench has a mass of 0.608 kg, and the astro- naut throws the wrench with a speed of 17.9 m/s.

How long does it take the astronaut to rea

To determine how long it takes for the astronaut to reach the spaceship, we can use the concept of conservation of momentum.

1. First, let's calculate the initial momentum of the astronaut-wrench system. Momentum (p) is given by the product of mass (m) and velocity (v):

Initial momentum of astronaut-wrench system:
P_initial = (mass of astronaut × velocity of astronaut) + (mass of wrench × velocity of wrench)

Given:
Mass of astronaut (m_astronaut) = 83.8 kg
Velocity of astronaut (v_astronaut) = 0 m/s (at rest relative to the spaceship)
Mass of wrench (m_wrench) = 0.608 kg
Velocity of wrench (v_wrench) = 17.9 m/s

P_initial = (83.8 kg × 0 m/s) + (0.608 kg × 17.9 m/s)
P_initial = 0 + 10.91812
P_initial ≈ 10.918 kg·m/s

2. Next, we need to calculate the final momentum of the astronaut-wrench system when the astronaut catches up to the spaceship. Since they are at rest relative to each other, the final velocity of the astronaut-wrench system is 0 m/s.

Final momentum of astronaut-wrench system:
P_final = (mass of astronaut + mass of wrench) × final velocity of astronaut-wrench system

Given:
Mass of astronaut (m_astronaut) = 83.8 kg
Mass of wrench (m_wrench) = 0.608 kg
Final velocity of astronaut-wrench system (v_final) = 0 m/s

P_final = (83.8 kg + 0.608 kg) × 0 m/s
P_final = 84.408 kg × 0 m/s
P_final = 0 kg·m/s

3. Based on the conservation of momentum principle, the initial momentum of the astronaut-wrench system is equal to the final momentum. Therefore:

P_initial = P_final

10.918 kg·m/s = 0 kg·m/s

4. Since the initial momentum is not equal to the final momentum, the astronaut will not be able to reach the spaceship without using a thruster.

To determine how long it takes for the astronaut to reach the spaceship, we need to calculate the time it takes for the astronaut to catch up to the spaceship after throwing the wrench.

First, we need to find the initial momentum of the astronaut and the wrench. The formula for momentum is:

momentum = mass * velocity

For the astronaut:
momentum_astronaut = mass_astronaut * velocity_astronaut

mass_astronaut = 83.8 kg
velocity_astronaut = 0 m/s (since the astronaut is at rest relative to the spaceship)

momentum_astronaut = 83.8 kg * 0 m/s = 0 kg·m/s

For the wrench:
momentum_wrench = mass_wrench * velocity_wrench

mass_wrench = 0.608 kg
velocity_wrench = 17.9 m/s

momentum_wrench = 0.608 kg * 17.9 m/s = 10.9328 kg·m/s

The total initial momentum of the system (astronaut + wrench) is conserved. Since the astronaut and the wrench are the only objects involved, the final momentum of the system after the wrench is thrown is equal to the initial momentum:

momentum_final = momentum_astronaut + momentum_wrench

Since the astronaut is now moving in the opposite direction to the initial velocity of the wrench, their momenta have opposite signs:

momentum_final = 0 kg·m/s - 10.9328 kg·m/s = -10.9328 kg·m/s

Now, let's calculate the velocity of the astronaut after throwing the wrench. Since momentum is equal to mass multiplied by velocity, we can rearrange the equation to solve for velocity:

velocity_final = momentum_final / mass_astronaut

velocity_final = -10.9328 kg·m/s / 83.8 kg ≈ -0.1307 m/s

The negative sign indicates that the astronaut is moving in the opposite direction to their initial direction of motion.

To calculate the time it takes for the astronaut to reach the spaceship, we'll use the equation:

velocity = displacement / time

Since the velocity and displacement are in opposite directions, we take the absolute value of the velocity:

velocity_final = -(39.6 m) / time

Solving for time:

time = -(39.6 m) / velocity_final = -(39.6 m) / (-0.1307 m/s)

time ≈ 302.6 seconds

Therefore, it takes approximately 302.6 seconds for the astronaut to reach the spaceship.