# What is the maximum speed that an object must reach to satisfy minimum transit time?

#### Davidllerenav

Hi, I need help with this problem:
1. The problem statement, all variables and given/known data
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?
2. Relevant equations
I guess this one:
$v= \frac {ds}{dt}$
3. 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?

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#### haruspex

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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."

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#### haruspex

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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

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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

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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

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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

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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

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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

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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 -a

You 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.

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#### 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

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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 ?

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#### haruspex

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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).

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#### 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

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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

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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.

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