Distance between rocket and object

[songoku]

Homework Statement

A rocket is launched upwards with initial speed of 120 m/s. After moving for 0.5 km, an object is released from the rocket. Given the rocket does not experience change in velocity, what is the distance between them after 10 seconds?

Relevant Equations

Conservation of momentum

Kinematics

Speed of rocket after moving 0.5 km = $\sqrt{u^2-2gh}=\sqrt{120^2 - 2 \times 9.81 \times 500}=3\sqrt{510}$ m/s

Then I try to consider conservation of momentum to find the speed of the object after being released.
Total momentum before the object is released = total momentum after the object released

Let:
m₁ = mass of rocket
m₂ = mass of object
v₁ = velocity of rocket after object being released = u (initial velocity) = $3\sqrt{510}$ m/s
v₂ = velocity of object after being released

So,
$$m_1.u + m_2.u=m_1.v_1+m_2.v_2$$
$$m_2.u=m_2.v_2$$
$$u=v_2$$

For momentum to be conserved, it means that after released the object will move upwards with the same speed as the rocket so the distance between them after 10 seconds will be zero?

Thanks

 

[Orodruin]

What about gravity?

 

[haruspex]

For momentum to be conserved

For momentum conservation of a system to apply, there must be no net external force on that system. See post #2.

 

[songoku]

What about gravity?

For momentum conservation of a system to apply, there must be no net external force on that system. See post #2.

Ah I see. So the rocket and the object is not closed system, Earth must be included for it to be closed system.

So for this question I just need to take the initial velocity of the object as $3\sqrt{510}$ m/s downwards?

Thanks

 

[Orodruin]

Ah I see. So the rocket and the object is not closed system, Earth must be included for it to be closed system.

So for this question I just need to take the initial velocity of the object as $3\sqrt{510}$ m/s downwards?

Thanks

No. How did you arrive at that? The object is following the rocket until it is released.

 

[songoku]

No. How did you arrive at that? The object is following the rocket until it is released.

I was just guessing.

Without guessing, my thought will be like this:
At the instant the object is released, it will have initial velocity which is the same as the rocket. Since the rocket is moving upwards, the initial velocity of the object is also upwards so the motion of the rocket and the object is the same. They will reach same maximum height at the same time so the distance between them will be zero

Thanks

 

[BvU]

Can the rocket maintain velocity in your scenario?

$\$

 

[haruspex]

the motion of the rocket and the object is the same.

"the rocket does not experience change in velocity"

 

[Orodruin]

I was just guessing.

Guessing without particular reason for the guess is usually not a good approach. (However, guessing based on accumulated experience can be very fruitful if you can later support it by computation.)

Since the rocket is moving upwards, the initial velocity of the object is also upwards so the motion of the rocket and the object is the same.

This is ok up until release.

They will reach same maximum height at the same time so the distance between them will be zero

No, the rocket by definition continues upwards at constant speed. Presumably due to the rocket engine countering gravity. After release, the released object has no such propulsion.

 

[songoku]

Ohhh I get it. Thank you very much for all the help and explanation Orodruin, haruspex, BvU

 

[Orodruin]

Ohhh I get it. Thank you very much for all the help and explanation Orodruin, haruspex, BvU

If you could post your solution, that would be helpful. In particular, it would help us ensure that you really did get it (not saying you didn’t, but many times people come back with similar misunderstandings after ”getting it”). It would also allow us to post alternative solutions.

 

[kuruman]

No, the rocket by definition continues upwards at constant speed. Presumably due to the rocket engine countering gravity. After release, the released object has no such propulsion.

If that is the case, why are we told in the statement of the problem that "A rocket is launched upwards with initial speed of 120 m/s."? Why specify "initial" and not "constant"? Another interpretation might be that the rocket is in free fall immediately after launch. After reaching 0.5 km height the object simply separates and the rocket experiences "no change" in whatever velocity it has at the height of release.

 

[BvU]

Another interpretation might be that the rocket is in free fall immediately after launch

@songoku: can the wording of the problem statement resolve the ambiguity ?

$\$

 

[Orodruin]

Another interpretation might be that the rocket is in free fall immediately after launch.

We are explicitly told that

Homework Statement:: ... Given the rocket does not experience change in velocity, ...

This seems completely incompatible with the rocket being in free fall.

 

[jbriggs444]

If that is the case, why are we told in the statement of the problem that "A rocket is launched upwards with initial speed of 120 m/s."? Why specify "initial" and not "constant"? Another interpretation might be that the rocket is in free fall immediately after launch. After reaching 0.5 km height the object simply separates and the rocket experiences "no change" in whatever velocity it has at the height of release.

The problem statement is intentionally worded to obfuscate things, for instance, by throwing in irrelevant numbers to [successfully in this case] distract the unwary student.

One can argue that this is fair play since real world problems have mounds of obfuscation and irrelevance to sift through.

 

[DaveC426913]

I am not sure how such homework problems are graded but is it acceptable - in the presence of ambiguity - to explicitly state an assumption and then solve it based on that? (For example, say "I interpret 'launched upward' to mean the presence of gravity.")
Or are you obliged to reach a specified answer?

 

[kuruman]

One can argue that this is fair play since real world problems have mounds of obfuscation and irrelevance to sift through.

One can also argue that, before one is deemed competent enough to attack real world problems, one first has to learn how the world can be modeled by tackling idealized problems in which there is no air resistance, cars going past each other are point masses, cows are spherical, etc. How can one separate the wheat from the chaff if one doesn't know what the wheat looks like? I believe that if the purpose of a problem is to ascertain whether a student understands the relevant basic principles and can apply them to answer a question, then deliberate obfuscation expressly defeats this purpose and turns homework problems into exercises in mind reading.

 

[kuruman]

I am not sure how such homework problems are graded but is it acceptable - in the presence of ambiguity - to explicitly state an assumption and then solve it based on that? (For example, say "I interpret 'launched upward' to mean the presence of gravity.")
Or are you obliged to reach a specified answer?

I think one is obliged to reach an answer that is consistent with what is given. It behooves the author of the question to ensure that no alternate answer can be reached consistently with the givens. I also think that allowing one to state one's assumptions is fraught with pitfalls.

For example, I could easily answer this rocket question by saying, "The problem does not mention gravity, therefore I assume that gravity is not present. Thus, when the object separates from the rocket, it will still be moving at 120 m/s and the distance between the two will be zero at all times including 10 s after separation."

 

[Orodruin]

For example, I could easily answer this rocket question by saying, "The problem does not mention gravity, therefore I assume that gravity is not present. Thus, when the object separates from the rocket, it will still be moving at 120 m/s and the distance between the two will be zero at all times including 10 s after separation."

Which is also the case if you assume the rocket to be in free fall in a gravitational field ... The existence or not of a gravitational field is irrelevant to the answer in that case.

 

[DaveC426913]

It behooves the author of the question to ensure that no alternate answer can be reached consistently with the givens.

Yes, but that's not what happened here. Hence my question.

 

[kuruman]

Yes, but that's not what happened here. Hence my question.

My opinion is that one must make the effort to answer the question as best as one can and not attempt to rewrite it. If there is perceived ambiguity (as in this case) one ought to raise one's hand and ask for clarification. However that works well in a class setting but not here. We, OP responders, often have to be interpreters and occasionally mind readers whether the obfuscation is deliberate or not. My personal belief is that questions in homework problems are meant to be parts of one's learning experience and that it's up to us to figure out what the question is trying to teach and reinforce it. Doing that can be challenging sometimes.

 

[haruspex]

The problem statement is intentionally worded to obfuscate things

Sometimes it turns out there are more parts to the question, and these explain the apparently irrelevant information. Is that the case here, @songoku?

 

[kuruman]

Which is also the case if you assume the rocket to be in free fall in a gravitational field ... The existence or not of a gravitational field is irrelevant to the answer in that case.

Exactly, but there is also the other case where gravity exists and the released object is affected by it but not the rocket because it presumably propelled by an engine against gravity at constant velocity.

 

[Orodruin]

Exactly, but there is also the other case where gravity exists and the released object is affected by it but not the rocket because it presumably propelled by an engine against gravity at constant velocity.

Sure, that is my original interpretation and it is still how I would read this problem. It is of course also possible that OP has not reproduced the exact problem statement word for word or that it is a translation where information was lost or altered.

 

[DaveC426913]

Having reread the problem with fresh eyes, I see no ambiguity. (What I do see is a contrived scenario, but that's de rigueur for homework problems.)

The problem says

• the rocket has an initial velocity upward of 120m/s,
• it remains 120m/s throughout the scenario,
• it reaches a height of .5km and releases an object,

How far apart are the rocket and object after 10 seconds?

There's really no ambiguity there.

• The altitude serves no purpose except to say "the object won't hit the ground".
• The rocket is moving upward (which means there is a "downward") for 10 seconds.
• while the object decelerates for 10 seconds.
• We can't discount gravity, or the answer degrades to a fairly meaningless "d=zero".
• And we have to discount air friction or the question would be intractable without more info.
• Without mass of rocket or object we don't have enough information to calculate anything about conservation of momentum. There is no compelling reason to assume the object has any significant mass.

There's two calculations there, one per body, that need to be added. That's seems just complex enough to give the student a challenge.

 

[Orodruin]

There's two calculations there, one per body

I disagree, there is just one computation needed, but that is all about solution strategy and the student’s understanding of the problem and underlying theory.

 

[DaveC426913]

I disagree, there is just one computation needed, but that is all about solution strategy and the student’s understanding of the problem and underlying theory.

Sure. You could do it as a single calc, or you calc them separately for clarity. Either way would be acceptable I think. Yours might be more elegant.

Regardless, the OP is def overthinking it.

 

[Orodruin]

Sure. You could do it as a single calc, or you calc them separately for clarity. Either way would be acceptable I think. Yours might be more elegant.

Regardless, the OP is def overthinking it.

I think there is significantly more clarity in the single calculation as it is also cleaner. Obviously, any correct solution is fine.

 

[songoku]

@songoku: can the wording of the problem statement resolve the ambiguity ?

$\$

Sorry I don't know, I have posted the exact wording of the question I got. I didn't alter anything from original question.

Sometimes it turns out there are more parts to the question, and these explain the apparently irrelevant information. Is that the case here, @songoku?

No, that's already the complete question, no other part.

turns homework problems into exercises in mind reading.

Yes, I really hate it when it happens

If you could post your solution, that would be helpful. In particular, it would help us ensure that you really did get it (not saying you didn’t, but many times people come back with similar misunderstandings after ”getting it”). It would also allow us to post alternative solutions.

Let: $u=120$ m/s

Distance between rocket and the object = distance traveled by rocket - distance traveled by object
$$=u.t-\left(u.t-\frac{1}{2} gt^2 \right)$$
$$=\frac{1}{2} gt^2$$
$$=\frac 1 2 (9.81)(100)$$
$$=491 ~\text{m}$$

 

[Orodruin]

As hinted above, I would have done the problem in the rocket’s rest frame, which would have gotten rid of the ut terms. Of course, it will end up with the same result.

 

[songoku]

Thank you very much for all the help and explanation Orodruin, haruspex, BvU, kuruman, jbriggs444, DaveC426913