A 87.2 kg astronaut is working on the engines of a spaceship that is drifting through space with a constant velocity. The astronaut turns away to look at Earth and several seconds later is 44.2 m behind the ship, at rest relative to the spaceship. The only way to return to the ship without a thruster is to throw a wrench directly away from the ship. The wrench has a mass of 0.411 kg, and the astronaut throws the wrench with a speed of 24.1 m/s.

How long does it take the astronaut to reachthe ship?
please show the steps

He has to close a gap of X = 44.2 m. Use conservation of momentumn to get his velocity relative to the ship, V, after he throws the wrench.

0.411*24.1 = 87.2 * V

The time required is X/V.

Sure, here are the steps to solve the problem:

1. Determine the initial momentum of the astronaut and the wrench system. Momentum (p) is given by the equation p = mass × velocity.

Momentum of the astronaut: p_astronaut = (87.2 kg) × 0 m/s (since the astronaut is at rest initially)
Momentum of the wrench: p_wrench = (0.411 kg) × (-24.1 m/s) (since the wrench is thrown away from the spaceship)

2. Apply the law of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event. In this case, the event is throwing the wrench away.

Total initial momentum = Total final momentum
(87.2 kg) × 0 m/s + (0.411 kg) × (-24.1 m/s) = (87.2 kg + 0.411 kg) × v_ship

Simplifying the equation: -9.9521 kg·m/s = 87.6112 kg × v_ship

3. Solve for the velocity of the spaceship (v_ship):

v_ship = -9.9521 kg·m/s / 87.6112 kg
v_ship ≈ -0.1136 m/s (Note: The negative sign indicates the motion is in the opposite direction to the initial motion)

4. Calculate the time taken by the astronaut to reach the ship using the equation of motion:

Distance = Initial velocity × Time + (1/2) × Acceleration × Time^2

In this case, the astronaut is at rest relative to the spaceship, so the initial velocity is 0 m/s, and the acceleration is also 0 m/s^2 (constant velocity).

Rearranging the equation: Time = Distance / Initial velocity
Time = 44.2 m / 0.1136 m/s
Time ≈ 388.14 s

So, it takes approximately 388.14 seconds for the astronaut to reach the ship.

To solve this problem, we can use the principle of conservation of momentum. The momentum before throwing the wrench would be equal to the momentum after throwing the wrench.

Step 1: Find the initial momentum of the astronaut and the wrench:

The initial momentum of the astronaut (before throwing the wrench) is given by:
P_astronaut = (mass_astronaut) × (velocity_astronaut)
P_astronaut = 87.2 kg × 0 m/s
P_astronaut = 0 kg·m/s

The initial momentum of the wrench is given by:
P_wrench = (mass_wrench) × (velocity_wrench)
P_wrench = 0.411 kg × (-24.1 m/s) (negative because the wrench is thrown away from the ship)
P_wrench = - 9.92251 kg·m/s (rounded to four decimal places)

Step 2: Find the final momentum of the astronaut and the wrench:

The final momentum of the astronaut and the wrench (after the astronaut reaches the ship) is zero since they are both at rest relative to the spaceship.

Step 3: Apply the conservation of momentum principle:

According to the conservation of momentum, the sum of the initial momenta should be equal to the sum of the final momenta:

P_initial = P_final
P_astronaut + P_wrench = 0

Substituting the values:
0 kg·m/s + (-9.92251 kg·m/s) = 0

Step 4: Find the time taken by the astronaut to reach the ship:

Since the spaceship is drifting through space with a constant velocity, the astronaut will take the same time to reach the ship as the wrench.

To find the time (t), we can use the equation of motion:

t = (final distance) / (final velocity)

The final distance is given as 44.2 m, and the final velocity is given as 0 m/s.

Substituting the values:
t = 44.2 m / 0 m/s

Since the denominator is zero, division by zero is undefined. Therefore, it is not possible to directly calculate the time taken for the astronaut to reach the ship using the given information.

Please double-check the given information or provide more information if available.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before throwing the wrench should be equal to the total momentum after throwing the wrench.

1. Identify the initial and final states:
- Initial state: The astronaut is at rest relative to the spaceship.
- Final state: The astronaut has reached the spaceship.

2. Determine the initial momentum:
The momentum of an object is given by the product of its mass and velocity (p = mv).
- Mass of the astronaut (m1) = 87.2 kg
- Velocity of the astronaut (v1) = 0 m/s
Therefore, the initial momentum of the astronaut is 0 (p1 = m1 * v1 = 87.2 kg * 0 m/s = 0 kg*m/s).

3. Determine the final momentum:
- Mass of the astronaut (m1) = 87.2 kg
- Velocity of the astronaut when they reach the spaceship (v2) = ?
Therefore, the final momentum of the astronaut is m1 * v2.

4. Determine the momentum change due to throwing the wrench:
- Mass of the wrench (m2) = 0.411 kg
- Initial velocity of the wrench (u2) = 0 m/s (since the wrench is initially at rest relative to the astronaut)
- Final velocity of the wrench (v2) = 24.1 m/s
Therefore, the momentum change of the wrench is given by Δp2 = m2 * (v2 - u2).

5. Apply the principle of conservation of momentum:
According to the principle of conservation of momentum, the initial momentum (p1) should be equal to the final momentum (p2).
p1 = p2
0 = m1 * v2 + Δp2

6. Solve for v2 (velocity of the astronaut when they reach the spaceship):
m1 * v2 = -Δp2
v2 = (-Δp2) / m1

7. Calculate Δp2 (momentum change of the wrench):
Δp2 = m2 * (v2 - u2) = 0.411 kg * (24.1 m/s - 0 m/s)

8. Substitute the values and calculate v2:
v2 = (-Δp2) / m1 = (-0.411 kg * (24.1 m/s)) / 87.2 kg

9. Calculate the time taken to reach the spaceship:
We can use the equation v = d / t to find the time (t).
t = d / v2
- Distance traveled by the astronaut (d) = 44.2 m
- Velocity of the astronaut when they reach the spaceship (v2) = calculated in step 8

10. Substitute the values and calculate t.

By following these steps, you can find the time it takes for the astronaut to reach the spaceship.