Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

Video transcript

Welcome back everybody. We are looking at two different moons of Uranus. Let me go ahead and draw out our moons here over just a little bit great. We are looking at titania And Oberon and we are told a couple different things about these two planets or Titania. We are told that the radius is 788. km. We're told that the mass of Titania is given by 35 two times on tier kilograms. And for Oberon, we are also given a radius of let's see here, 761. kilometers and a mass of 30.1 times. And to the 20 kg. Now we are asked a couple of things, we are asked to say we were to have some object that were to escape the atmosphere of both of these planets. What is going to be the escape velocity of those objects for each planet. And then we are also asked why the escape speed is independent of the escaping objects mass. So let's go ahead and establish or try to figure out some equations that we can use here. I'm seeing that we're looking at radius is and masses and velocities. The first thing that comes to my mind is the energy conservation equation that states that the initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy. Let's plug in some terms here, our initial kinetic energy will be given by one half times the mass of our escaping object, times our initial velocity squared uh minus because we're looking at the potential energy of spheres really big G, which is just some constant for planet, times the mass of our escaping object, times the mass of our big planet of our planets. All over the radius of our planets. Now, what is this equal to? What are we looking on the right side of this equation? Well, these objects are gonna travel forever, so both of these at infinity, simply just go to zero. The potential energy, it's not really gonna be connected anymore. And the kinetic energy, it's not you're not really gonna have the influence of those planets. The right side of this equation is just going to be zero. This gives us if we add this term right here to both sides, we have that one half times the mass of our escaping objects, times our escape velocity is equal to our G. Constant times little M times big M. All over our radius. Now we want this escape velocity right here. So I'm actually gonna rearrange the equation to isolate it. And we get that our escape velocity for each planet is equal to the square root of two times big G. Time, times big M. All over the radius of our planet's the little M's here canceled out. So now that we have the right equation, let's go ahead and do that for each planet. Right? So the escape velocity for our object from titania going to be equal to square root. And let's we know all these values. So let's just plug them in here, Right, two times big G. Which is just a constant equal to 6.67 times 10 to the negative 11. And this radical here times the mass of our planet titania, which is 35.2 times 10 to all over our radius of our planet, which is 788 kilometers or 788,900 m. When you plug this into your calculator, we get that. The escape velocity from planet titania is we're sorry, the moon titania 771.5 m per second. Easy enough. We repeat all these steps now and we are looking for the escape velocity of this object from Oberon. So once again, let's just plug in all the values we know two times are big, constant Times the mass of Oberon given by 30.1 times 10 to the kg. All divided by our radius of. Let's see here, 761 km or 1004 100 m. Which when you plug this into your calculator, you get that the escape velocity for Oberon is 726.2 m per second. So now let's look at this. Why is the escape speed independent of the escaping objects mass. Well, let's take a look here when we were looking at our energy conservation equation. If you notice when I was plugging those initial values, the mass of the escaping object is taken into consideration for both the kinetic energy and potential energy, meaning that the kinetic and potential energy are proportional to mass are proportional to the mass. So now that we have found our escape velocities, and we know that kinetic and potential energy are proportional to mass, this gives us a final answer. Choice of D. Thank you all so much for watching hope. This video helped. We will see you all in the next one.