Max Speed to Move from Point A to Point B

[Davidllerenav]

Hi, I need help with this problem:

Homework Statement

Condition: an object has to move from point A to point B in the least time possible. The distance between the points is L. The object can accelerate (decelerate) with a fixed acceleration $a$ or move with a constant speed.

What maximum speed does this object have to reach to satisfy the condition?

Homework Equations

I guess this one:
$v= \frac {ds}{dt}$

The Attempt at a Solution

If the object moves from A to B in the least time possible, that means that $\Delta t$ tends to $0$. To find the maximum speed I need to find the moment when the object travels more dinstance in the least time. That would be the derivative of the distance $L$ with respect to $t$, so $lim_{\Delta t\to 0} \frac{\Delta L} {\Delta t} = \frac {dL} {dt}$. Am I right? The problem is that I don't know hot to find the maximum speed necessary. How do I find the maximum speed?

 

[haruspex]

The object can accelerate (decelerate) with a fixed acceleration to or move with a constant speed.

The question makes no sense to me. Is this the exact wording? Is it a translation?

 

[Davidllerenav]

The question makes no sense to me. Is this the exact wording? Is it a translation?

It is a translation. It is $a$ instead of "to". So it would be "The object can accelerate (decelerate) with a fixed acceleration ##a## or move with a constant speed."

 

[haruspex]

It is a translation. It is $a$ instead of "to". So it would be "The object can accelerate (decelerate) with a fixed acceleration ##a## or move with a constant speed."

Any constant speed or a given one? Is relativity to be taken into account?

 

[Davidllerenav]

Any constant speed or a given one? Is relativity to be taken into account?

No costant speed. What do you mean with relativity?

 

[haruspex]

No costant speed. What do you mean with relativity?

Are we limited to the speed of light?

 

[Davidllerenav]

Are we limited to the speed of light?

Relativity is definitely out of the picture here.

 

[haruspex]

No costant speed.

Sorry, I do not understand that answer. We are told we can use the constant acceleration, a, or a constant speed. Or does it mean the object can switch between the two, having started from rest?

 

[haruspex]

Yes. Relativity is definitely out of the picture here.

If relativity is not assumed then the answer is no, we are not limited to the speed of light.

 

[Davidllerenav]

Sorry, I do not understand that answer. We are told we can use the constant acceleration, a, or a constant speed. Or does it mean the object can switch between the two, having started from rest?

That's a good question. I guess that it meas that we can use the constant acceleration or the constant speed, since it doesn't say anithing about it starting from rest.

 

[haruspex]

That's a good question. I guess that it meas that we can use the constant acceleration or the constant speed, since it doesn't say anithing about it starting from rest.

I'm struggling to find an interpretation that makes for a reasonable question. It seems obvious just to say go at an infinite speed.

If the choice is between
(1) a given acceleration, a, starting from rest and
(2) a given constant speed v
then it will depend on the numeric relationship between a, v and L.

Maybe you have to start and finish at rest, and you can mix accelerating at a, decelerating at a, and moving at constant speed.

 

[Davidllerenav]

I'm struggling to find an interpretation that makes for a reasonable question. It seems obvious just to say go at an infinite speed.

Why is it obvious? Wouldn't it be a non-constant speed then?

If the choice is between
(1) a given acceleration, a, starting from rest and
(2) a given constant speed v
then it will depend on the numeric relationship between a, v and L.

Maybe you have to start and finish at rest, and you can mix accelerating at a, decelerating at a, and moving at constant speed.

I really don't know, I guess that it can start at rest, but why woudl it finish at rest?

 

[haruspex]

Why is it obvious? Wouldn't it be a non-constant speed then?

If it does not have to start at rest then you can start at the dedired speed. If you rule out an infinite speed there is no answer, because whatever speed you pick you can do better by going faster.

why woudl it finish at rest?

I am suggesting that as a requirement in order to make it a reasonable question.

 

[CWatters]

I agree. Question as written is too obvious. To turn it into a sensible question you must assume it starts or finishes at rest and its most likely they meant you to assume both. So the trip must be divided into three parts...

Acceleration at a
Constant velocity
Deceleration at -aYou need to work out how long each part should be to make the trip in the shortest time, and write an equation for the Max velocity achieved.

 

[Davidllerenav]

If it does not have to start at rest then you can start at the dedired speed. If you rule out an infinite speed there is no answer, because whatever speed you pick you can do better by going faster.

I am suggesting that as a requirement in order to make it a reasonable question.

What about if relativity is considered? If the object can't go with infinite speed.

 

[hmmm27]

I would suggest clarification from the teacher. Like, lots of clarification and highly suggested.

Also, is the answer to be found using calculus? or would simple kinematic equations do.

 

[Davidllerenav]

I would suggest clarification from the teacher. Like, lots of clarification and highly suggested.

Also, is the answer to be found using calculus? or would simple kinematic equations do.

I guess kinematics. Since the last topic we saw was Galilean transformation.

 

[haruspex]

Also, is the answer to be found using calculus? or would simple kinematic equations do.

Whatever the intent, calculus would be overkill.
And you mean kinetics; kinematics is something else.

 

[hmmm27]

Whatever the intent, calculus would be overkill.

Unless it was a math, not physics, class.

And you mean kinetics; kinematics is something else.

LOL, I was wondering about that the other day : apologies for adding to existing confusion.

Wait, what ?

 

[haruspex]

Unless it was a math, not physics, class.

LOL, I was wondering about that the other day : apologies for adding to existing confusion.

Wait, what ?

Kinematics is not concerned with masses or forces. It is sometimes described as the geometry of motion. E.g. in a mechanical linkage it addresses how the components are constrained to move in relation to each other.

 

[hmmm27]

The OP's question appears to be easily solveable (once the missing bits are found) with "kinematics equations", found in Wikipedia or our own reference post : time, distance, velocity (and acceleration).

 

[Davidllerenav]

Kinematics is not concerned with masses or forces. It is sometimes described as the geometry of motion. E.g. in a mechanical linkage it addresses how the components are constrained to move in relation to each other.

But I think that we need to use kinematics, because we haven't see dynamics yet. Kinectics and dynamics are the same, right?

 

[haruspex]

But I think that we need to use kinematics, because we haven't see dynamics yet. Kinectics and dynamics are the same, right?

I erred.
I am used to people referring to kinematics in the context of questions involving accelerations resulting from torques and forces. In the present case, we are unconcerned with such - it is purely a question of the relationships between position, velocity, acceleration and time.

 

[Davidllerenav]

I erred.
I am used to people referring to kinematics in the context of questions involving accelerations resulting from torques and forces. In the present case, we are unconcerned with such - it is purely a question of the relationships between position, velocity, acceleration and time.

Exactly. So, If the answer would be infinite speed with no relativity. If now we are limited by speed of light, what would be the answer?

 

[haruspex]

Exactly. So, If the answer would be infinite speed with no relativity. If now we are limited by speed of light, what would be the answer?

I think you can make a stab at that.

 

[Davidllerenav]

I think you can make a stab at that.

I guess that it would be when it is near B, because it is under constant acceleration.

 

[haruspex]

I guess that it would be when it is near B, because it is under constant acceleration.

When what is near B? When A is near B? A and B are given, you can't move them.
It is a very simple question - if you want to get there ASAP, and your speed is limited only by the speed of light, what speed do you have to go at (as near as possible)?

 

[Davidllerenav]

When what is near B? When A is near B? A and B are given, you can't move them.
It is a very simple question - if you want to get there ASAP, and your speed is limited only by the speed of light, what speed do you have to go at (as near as possible)?

When the object moving from A to B is near B, because it is under a constant acceleration.
To answer your question, it would be as near as the speed of light.

 

[haruspex]

When the object moving from A to B is near B, because it is under a constant acceleration.

Sorry, but I have no idea what you are trying to say.

To answer your question, it would be as near as [possible to] the speed of light.

Yes.

 

[Davidllerenav]

Sorry, but I have no idea what you are trying to say.

I'll try to explain myself better. The objecc is moving from A to B, starting at rest with a constant acceleration a until it reaches B. So the fastest the object will be traveling would be when it gets to point B.

 

[haruspex]

The objecc is moving from A to B, starting at rest with a constant acceleration a

My comment about relativity and infinite speed was in the context of the constant speed case.
For constant acceleration it gets a bit tricky. Not sure what that means within relativity.

 

[Davidllerenav]

My comment about relativity and infinite speed was in the context of the constant speed case.
For constant acceleration it gets a bit tricky. Not sure what that means within relativity.

So, if in the case of constant speed, the object speed must be near the speed of light.

 

[haruspex]

So, if in the case of constant speed, the object speed must be near the speed of light.

Yes, depending on what exactly the question was intended to say.

 

[Davidllerenav]

Yes, depending on what exactly the question was intended to say.

I really don't know that the questio was trying to say. So well, I guess that there arte those two cases.

 

[CWatters]

You seem to be making this problem much harder than it is. I very much doubt they intended you to think about limitations due to the speed of light!

It starts from A at rest and accelerates at "a". At some point it must stop accelerating and start to decelerate at "a" in order to be at rest at B.

A few basic equations of motion (eg SUVAT) and you are done.