You and your team get a distress call from another ship...

You and your team get a distress call from another ship indicating that there has been an explosion and their propulsion and navigation systems are now off line. A large piece of their ship has been jettisoned into space, and their life support systems are failing. They need help as soon as possible. You try to track them, but are only able to pick up the piece of the craft that had been projected away from them in the explosion. You look up the make and model of their ship, a Galaxy Explorer 5000 (GE 5000 ), which has a total mass of 55000 kg. The piece heading in your direction is the ship's storage module, and has a mass of 7500 kg. The GE 5000 was at rest with respect to your ship at the time of the explosion. (a) Assuming you received the distress call 20 minutes before the jettisoned storage module passed by your ship at a speed of $750.0 \mathrm{m} / \mathrm{s},$ what is the speed of the damaged craft? (b) How far away is the Galaxy Explorer 5000 when its storage module passes by? (c) In which direction, and how fast must you go, in order to catch up with the GE 5000 before its life support systems fail. (The crew had 80 minutes of life support left when they made the distress call, so you have 60 minutes to get to them.)

You and your team get a distress call from another ship indicating that there has been an explosion and their propulsion and navigation systems are now off line. A large piece of their ship has been jettisoned into space, and their life support systems are failing. They need help as soon as possible. You try to track them, but are only able to pick up the piece of the craft that had been projected away from them in the explosion. You look up the make and model of their ship, a Galaxy Explorer 5000 (GE 5000 ), which has a total mass of 55000 kg. The piece heading in your direction is the ship's storage module, and has a mass of 7500 kg. The GE 5000 was at rest with respect to your ship at the time of the explosion.
(a) Assuming you received the distress call 20 minutes before the jettisoned storage module passed by your ship at a speed of $750.0 \mathrm{m} / \mathrm{s},$ what is the speed of the damaged craft? (b) How far away is the Galaxy Explorer 5000 when its storage module passes by? (c) In which direction, and how fast must you go, in order to catch up with the GE 5000 before its life support systems fail. (The crew had 80 minutes of life support left when they made the distress call, so you have 60 minutes to get to them.)

Key Concepts

Conservation of Momentum

In any explosion or separation event in space—where external forces are negligible—the total momentum of the system remains constant. This principle allows us to relate the velocities and masses of the separated components immediately after the explosion and is fundamental to understanding how momentum is redistributed among fragments.

Reference Frames and Relative Velocity

When analyzing motions in space, it is essential to define a reference frame. Relative velocity refers to the speed of an object as seen from another moving object. In this problem, knowing the speeds relative to each other helps calculate the true speeds and directions necessary for an intercept mission.

Kinematic Equations and Time-Distance Relationships

Calculating how far an object has traveled over a certain period using its velocity is crucial in determining distances in space. The simple relation distance = speed × time underlines much of the analysis in intercept scenarios, where timing is critical to meet mission deadlines or safety limits.

Interception Course and Pursuit Strategy

Planning an interception involves determining the proper heading and required speed to close the gap between two moving objects within a limited time. This concept combines relative motion analysis, kinematics, and the initial conditions post-event to create an effective pursuit strategy ensuring timely rescue or engagement.

Transcript

00:01 Hello.

00:02 In part a of this problem, we are needing to use the information we gained from the storage module that is passing by our ship to figure out the speed of the ship that is in distress.

00:18 So we are going to use conservation of momentum, where we know that the momentum of the ship before the explosion was at rest with respect to a moment.

00:32 Our ship.

00:33 So the total momentum of both things needs to be zero meters per second, which means that the momentum of the ship plus the momentum of the jettisoned storage module need to be equal and opposite to each other.

00:49 So they're going to be moving in opposite directions.

00:52 Now, i made the assumption that the storage module's mass is separate from the mass of the ship because it was a little uncomfortable.

01:02 Clear whether or not that was included in the total mass of the ship.

01:06 So i'm going to make the assumption that it is a separate storage module.

01:12 But if it should be included as part of the mass of the ship, then m1 would just be 7 ,500 kilograms less.

01:21 So be dropped down to 48, 48500.

01:27 So let's set these two equal to each other.

01:30 Momentum 1 is the momentum of our spacecraft that we're hoping to find.

01:37 So that's m1 times v1.

01:39 And that's going to be equal to, and in the opposite direction of m2, which is the storage module that is headed up.

01:50 And we know m1, we know m2, we know v2.

01:53 So we just need that v1 component.

01:56 So we divide both sides by m1.

01:58 And we're going to get this equation right here.

02:03 If we plug our numbers into this equation, we should get a value of 102 .3 rounding to the nearest 10 meters per second.

02:18 This is going to be in the opposite direction that the module is, or the storage module is headed.

02:30 Part b just asks us to figure out how far away this ship is at this time.

02:37 So we are told that 20 minutes has passed and that the module, the storage module is traveling at 750 meters per second.

02:50 And we also know that the actual ship is moving at this speed now, 102 .3.

02:56 So the distance that the ship is away from us is going to be equal to the distance that the storage module traveled plus the distance that it traveled.

03:10 So this is the total distance away from us that this thing is.

03:15 Well, the distance that the storage module traveled is going to be equal to its speed.

03:20 Sorry, this is the storage module.

03:23 Two times the time that it has been moving.

03:28 And the distance that the ship itself has traveled is going to be its speed times the time...

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