Hey, guys. Let's work out this problem together. So we're landing on the surface of a large spherical asteroid. We're going to set off walking in one direction, blah, blah, blah. We're going to drop some tools. And eventually, we have to figure out how fast do we have to jump in order to escape this asteroid. So what variable is that? What variable are we talking about here? We're talking about an escape velocity. So we can look at our escape velocity formula. So we've got vescape is equal to 2GM/r. If I take a look at this equation, I need to figure out the escape velocity. I know that G is just a constant. I do not have the mass of the asteroid, and I also don't have the initial distance. I don't have any information about that. So this is a problem where we're going to have to basically go in a different direction and get what these variables are. We're going to have to go and get M and then go and get little r and then plug them back into this problem. I like this problem because it's going to combine a lot of different things from this chapter.
Let's first start out with just drawing a diagram. So I've got a large spherical asteroid that I am going to land on, like that. And the first part of this problem says that I'm going to set off walking in one direction, and after some later time, I've realized that I've basically come back around to, return to my spaceship. And I checked my little pedometer on my watch, and I've gone 25 kilometers. So that first part of the problem means that we are traveling around a spherical asteroid, which means that we're basically traveling around the circumference of it. This represents the circumference. And let's check out what that circumference means in terms of an equation. We know that the circumference of a circle is related to the radius of that circle by C=2πr. So we can actually go ahead and figure out what the radius of this circle is, this little r distance, by using this equation, C=2πr. We know what the circumference is. That's 25,000 in terms of meters. All we have to do is divide it by 2π, and we get the radius. And that's about 3,980 meters. So that is one part. So now we have the radius.
Now I need to go ahead and figure out what the mass is, and I'm going to do that over here somewhere. So I need to figure out what the mass of this asteroid is. So we're done with this first part of the problem. Now we've got to look at the second part. We're going to grab a tool from the toolbox and drop it, and then noting that, using our watch or something like that, it takes about 30 seconds to hit the ground from 1.4 meters high. So if we're looking for the big mass of the planet, we could use forces, gravitational forces, or gravitational accelerations. This part of the problem right here, this dropping 30 seconds and hitting the ground, looks like a kinematics problem, vertical kinematics, which we've done a lot of before. So I want to just go ahead and draw a quick diagram of what's going on in that part. So we've got this little, this ground right here, and I'm going to be dropping a rock, and I'm told that the distance that it falls is equal to 1.4. It takes a time of 30 seconds, and we know that it is dropped, which means that the initial velocity is equal to 0. So what is my target variable here? Well, I'm looking for, let's see. If I can keep going, this is going to be my Δy, right? It's going to be the distance this thing falls. I have the final velocity when it gets to the surface, but I don't know what that is. And I have the gravitational acceleration. That is going to be g at the surface. Right? Because this is going to be a gravitational acceleration, but we're standing really close to the surface of the asteroid. And I also don't know what that is. So if I take a look at my equations for M, my M equations involve forces and gravitational accelerations. Forces, I don't have any information about masses and forces, but I do have some sort of kinematics in which I have this acceleration. So I'm going to grab gsurface as the equation I'm going to use, because I'm on the surface, and that's going to be GM/r2. So now what happens is if I can figure out what this mass is, let me go ahead and rearrange for that. Right? So I've got gsurface,r2/G=M. So I just need to figure out what this gsurface is by looking at this part of the problem. So I basically got to go somewhere else to go and find out what gsurface is. Let's go ahead and do that. So if I'm looking for gsurface, I need that's a kinematic variable. I need 3 of the 5 other ones. Right? So I need, I've got the velocity initial, velocity final. I've got the t and the Δy. Which one of those do I which ones which one of these variables do I have? I have the initial velocity. I don't have the final velocity. I have the time and I have the Δy. So I'm going to pick the equation that doesn't have that vfinal. Hopefully, you guys remember your kinematic formulas. This is going to be Δy equals initial velocity t+12gt2. Now, we know that the initial velocity is equal to 0, so that means that this is going to cancel out. And then we basically can choose this downward direction to be positive, because then everything turns out to be positive. So the Δy is going to be 1.4, that's the meters. And then we've got one-half of g, and this is going to be gsurface, and then the t is just equal to 30, and then we have that squared. So let me go ahead and move down. So basically, I'm going to move this one half over to the other side and then the 30 squared goes down. So we basically end up with 2 times 1.4 divided by 30 squared. Let me go ahead and reiterate that. Divided by 30 squared is equal to gsurface. So you go ahead and plug all that stuff into your calculator. You should get that gsurface=3.11×10-3. So now we have gsurface. So now we can plug this thing back into this equation. And if we go ahead and solve for M, we're going to get 3.11×10tothe-3,3,9802,dividedby6.67×10-11.G. And that's going to be the mass of the spherical asteroid, which is equal to We've got 7.39×1014. That's in kilograms. Now that I have this mass right here, I can plug this all the way back into that equation, and I can figure out what the final escape velocity is going to be. So it's basically just like working outwards to find all your variables and then plugging them all back into your original equation. 7×10tothe-11. Now I've got the mass of the asteroid, which is, 7.39×1014. So let me go ahead and actually move all of this stuff down. I'm going to move this stuff down a little bit. There we go. And then I've got, divided by r, the distance, which is going to be 3,980. You go ahead and plug all this stuff in. You're going to get the escape velocity is equal to 4.98. That's meters per second. So this is how fast you'd have to be jumping, which is actually probably doable if you were actually, like, in a spacesuit. So this is the final answer for how fast you'd have to be traveling to escape this asteroid. Let me know if you guys have any questions.