A \(65-\mathrm{kg}\) astronaut is floating in empty space. If she shines a 1.0-W flashlight in a fixed direction, how long will it take her to accelerate to \(10 \mathrm{m} / \mathrm{s} ?\)

An astronaut of mass \(65 kg\) is shining a flashlight of power \(1.0 W\) in a fixed direction. We are to calculate how long it will take for the astronaut to accelerate to \(10 m/s\). The speed of light is \(3 \times 10^8 m/s\). By the conservation of momentum, the momentum of the astronaut must equal to the momentum of the photons shone by the flashlight.

The rate of change of momentum for the flashlight equals the rate of change of momentum for the astronaut in magnitude. This can be formulated as follows: \(P_{flashlight} = vP_{astronaut}\). Now \(P_{flashlight}\), the power of the flashlight, is the energy emitted per unit time, so it can also be written as the energy of each photon times the number of photons emitted per unit time, which gives us: \(P_{flashlight} = hf_{photon}\). Also \(P_{astronaut} = ma\) can be written as the rate of change of momentum for the astronaut. Here, \(v\) is the velocity of light, \(h\) is Planck’s constant, \(f_{photon}\) is the frequency of the light being emitted, \(m\) is the mass of the astronaut and \(a\) is her acceleration.

Assuming that the light being emitted by the flashlight is in the visible spectrum, its energy can be approximated by using \(E = hf_{photon} = p_{photon}v\), where \(p_{photon}\) is the momentum of a single photon. By substituting this into our first equation we get a new equation linking acceleration and time as \[t = \frac{mv_{f}}{P_{flashlight}}\], where \(v_{f}\) is the final velocity of the astronaut.

We can now substitute the given values into this equation to find the time. This gives us: \(t = \frac{(65 kg)(10 m/s)}{1 W} = 650 s\).