⋯Light Rocket Stranded 12 m from your spacecraft, you realize that your flashlight makes a direc

⋯Light Rocket Stranded 12 m from your spacecraft, you...

$\cdots$ Light Rocket Stranded 12 $\mathrm{m}$ from your spacecraft, you realize that your flashlight makes a directed beam of intensity 950 $\mathrm{W} / \mathrm{m}^{2}$ and can be used to propel you back to safety. (a) If the radius of the beam is $3.8 \mathrm{cm},$ how much force does it exert on you? (b) If your mass (with spacesuit and gear) is 95 $\mathrm{kg}$ , what acceleration does the flashlight give you? (c) How much time will it take for you to reach the spacecraft? (d) Frustrated by the slow pace, you throw the 0.82 -kg flashlight at 22 $\mathrm{m} / \mathrm{s}$ directly away from the spacecraft. In how much time will you reach the spacecraft after you throw the flashlight?

$\cdots$ Light Rocket Stranded 12 $\mathrm{m}$ from your spacecraft, you realize
that your flashlight makes a directed beam of intensity 950 $\mathrm{W} / \mathrm{m}^{2}$
and can be used to propel you back to safety. (a) If the radius of
the beam is $3.8 \mathrm{cm},$ how much force does it exert on you? (b) If
your mass (with spacesuit and gear) is 95 $\mathrm{kg}$ , what acceleration
does the flashlight give you? (c) How much time will it take for
you to reach the spacecraft? (d) Frustrated by the slow pace, you
throw the 0.82 -kg flashlight at 22 $\mathrm{m} / \mathrm{s}$ directly away from the
spacecraft. In how much time will you reach the spacecraft after
you throw the flashlight?

Key Concepts

Conservation of Momentum

This principle states that in an isolated system, the total momentum remains constant if no external forces act on it. This is important when considering actions like throwing an object, as ejecting mass in one direction results in a recoil or change in momentum of the remaining mass in the opposite direction.

Radiation Pressure

This concept explains that electromagnetic radiation carries momentum and exerts pressure on surfaces upon which it impinges. The pressure is linked to the light’s intensity and the speed of light, and it is responsible for transferring momentum from photons to matter. In many cases, the radiation pressure is calculated using the relation P = I/c for absorbing surfaces.

Force from Light

A directed beam of light can exert force when its momentum is transferred to an object. The force is determined by multiplying the radiation pressure by the cross?sectional area of the beam. This allows for a calculation where F = (I * A) / c, linking the beam's intensity, the illuminated area, and the speed of light.

Newton’s Second Law

Newton’s second law establishes the relationship between force, mass, and acceleration, expressed as F = ma. It is used to translate the force from the light beam into the acceleration imparted to an object, which is crucial for understanding how the object’s motion changes over time.

Kinematics of Constant Acceleration

When an object experiences a constant acceleration, its motion is described by kinematic equations that relate displacement, velocity, acceleration, and time. These equations enable the calculation of how long it takes an object to travel a specified distance under the influence of a steady acceleration.

Transcript

00:01 Okay, so as an astronaut, as a stranded astronaut, you can proper yourself by either switching on the flashlight, producing a radiation force, or by throwing the flashlight directly opposite the direction we wish to travel.

00:21 So the first item of this problem, we just need to calculate the radiation force, the radiation force of the flashlight.

00:32 So this is just the pressure multiply by the surface area.

00:45 And the pressure, as we know, is just the intensity divided by the speed of light.

00:52 And the surface area is a circle, which is pi r square.

01:00 So calculating this force, we have intensity, which is 950.

01:07 Divided by the speed of light just 3 times 10 to the 8 multiplies by pi 0 .0338 square.

01:27 Calculating this we have 1 .44 times 10 to the minus 8 newtons and that's the radiance force now we have to use the newton second law using the newton second law to the second item of this problem to calculate the acceleration so the acceleration is just the force the radiation force that we calculate the radiation force divided by the mass of the astronaut so this is just 1 .44 times 10 to the 8 divided by 95 kilograms, which gives us 1 .5 times 10 to the minus 10 meters per second square.

02:46 This is the acceleration produced by the flashlight.

02:51 So let's see.

02:54 Now we have to calculate the time required to return to the spacecraft.

03:05 So to calculate this, just remember of the equation for the displacement, which is v0 v0 t plus a t squared divided by 2, for which in this case is the initial velocity is 0.

03:34 So we can say that the time is just two times the displacement divided by the acceleration and the square...

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