An 80 -kg astronaut becomes separated from his spaceship. He is \(15.0 \mathrm{~m}\) away from it and at rest relative to it. In an effort to get back, he throws a \(500-\mathrm{g}\) object with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) in a direction away from the ship. How long does it take him to get back to the ship? a) \(1 \mathrm{~s}\) b) \(10 \mathrm{~s}\) c) \(20 \mathrm{~s}\) d) \(200 \mathrm{~s}\) e) \(300 \mathrm{~s}\)

In this problem, we have the following given values: - Mass of astronaut (M): 80 kg - Mass of the object (m): 500 g = 0.5 kg (converted to kg) - Initial distance from the ship (d): 15.0 m - Velocity of the thrown object (v): 8.0 m/s

According to the conservation of momentum, the momentum before throwing the object is equal to the momentum after throwing it away. Let's represent the astronaut's initial velocity as V_i and final velocity as V_f, and the final velocity of the object as v_f. Since both astronaut and object are initially at rest, their initial velocities are 0. Initially, Total momentum = M * V_i + m * V_i = 0 (because V_i = 0) After throwing, Total momentum = M * V_f + m * v_f By conservation of momentum, these two momenta are equal: 0 = M * V_f + m * v_f Now, we can solve this equation to find V_f, the final velocity of the astronaut.

We have the equation 0 = M * V_f + m * v_f. Now we can substitute the given values and solve for V_f. 0 = (80 kg) * V_f + (0.5 kg) * (8.0 m/s) Rearranging the terms to isolate V_f: V_f = - (0.5 kg * 8.0 m/s) / 80 kg Now, we can calculate V_f: V_f = - (4.0 kg * m/s) / 80 kg = -0.05 m/s The negative sign means the astronaut will move towards the spaceship, which is the desired direction.

Now that we have the final velocity of the astronaut (V_f = -0.05 m/s), we can use it to find the time (t) it will take for him to cover the 15.0 m distance back to the ship. Using the linear motion equation with constant velocity: distance = velocity * time We can rearrange this equation to find the time: t = distance / velocity Now we substitute the known values: t = 15.0 m / -0.05 m/s The negative sign in the velocity will cancel out, because the distance is given in positive values and astronaut is moving towards the spaceship, as expected. Thus, we calculate the time as: t = 15.0 m / 0.05 m/s = 300 s Thus, it takes the astronaut 300 seconds to reach the spaceship. The correct answer is: e) \(300\,\text{s}\).