A race between two objects with the same speed?

[LM542]

Two objects travel from point A to B. They start with the same speed.

1. The first object travels without any obstacles from A to B.

2. The second object encounters at the middle of the course a planet without an athmosphere. The planet has a tunnel (don't ask why) through which the object flies. The object has enough escape velocity.

The Gravity of the Planet should increase the speed of the object and when the object tries to escape, the speed will decrease.

Will both objects reach the finish line at the same moment?
The first object travels with constant speed, does the second object ever fall below the speed of the first object?

 

[rootone]

The object passing through the imaginary tunnel planet would at first increase in speed relative to the the other object due to being accelerated by the planet's gravity.
Having passed through the tunnel, it would then decelerate by the same amount and the net effect should be that they both arrive at the destination simultaneously.
(Assuming no other bodies are present in the system, and we are not talking about relativistic kinds of speed.)

 

[berkeman]

Two objects travel from point A to B. They start with the same speed.

1. The first object travels without any obstacles from A to B.

2. The second object encounters at the middle of the course a planet without an athmosphere. The planet has a tunnel (don't ask why) through which the object flies. The object has enough escape velocity.

The Gravity of the Planet should increase the speed of the object and when the object tries to escape, the speed will decrease.

Will both objects reach the finish line at the same moment?
The first object travels with constant speed, does the second object ever fall below the speed of the first object?

What do you think? Draw a sketch of v(t) for both objects on the same graph. What can you say about the average speed of each?

 

[LM542]

They object passing through the imaginary tunnel planet would at first increase in speed relative to the the other object due to being accelerated by the planet's gravity.
Having passed through the tunnel, it would then decelerate by the same amount and the net effect should be that they both arrive at the destination simultaneously.
(Assuming no other bodies are present in the system, and we are not talking about relativistic kinds of speed.)

A better analogy:
There are two runners who run together at 10 kmh, one of them suddenly accelerates to 20 kmh and then reduces to 10 kmh. The both continue to run with 10 kmh to the finish line. Will both come at the same time to the finish line. The first ran exactly at 10 kmh, the second ran almost the whole time at 10 kmh but had a sudden acceleration. They can't reach the finish line at the same time. The second runner has to reduce his speed below 10 kmh that both can reach the finish line together.Does the second object ever travel below the speed of the first object, otherwise they can't reach the finish line at the same time.

 

[rootone]

The two runners are not an equivalent scenario to that of the two traveling bodies which you described.
In the case of the runners, one of them has applied an additional amount of energy and will therefore arrive before the runner who did not.
In the case of the traveling objects through space, no additional energy is added in total.
The object going through the tunnel will arrive at the destination at a very slightly slower velocity because it still is subject, however minutely, to the gravitational field of the planet.
It will be traveling slower than the the unaffected object by exactly the same amount that it has to be so that they arrive at the same time since no additional energy was applied overall.

Another consideratation is this though.
So far I have been visualising your tunnel planet as being half way between then starting and ending points.
If it was in fact closer to the ending point, your object would would get much more acceleration from it than deceleration before it reaches the destination.
It would arrive sooner than the unaffected object.

 

[phyzguy]

The object passing through the imaginary tunnel planet would at first increase in speed relative to the the other object due to being accelerated by the planet's gravity.
Having passed through the tunnel, it would then decelerate by the same amount and the net effect should be that they both arrive at the destination simultaneously.
(Assuming no other bodies are present in the system, and we are not talking about relativistic kinds of speed.)

Think about what you said. Object A goes at constant speed. Object B starts at the same speed, accelerates to faster than A, then decelerates back to the same speed as A. Do you really think they will arrive at the same time?

 

[LM542]

The object going through the tunnel will arrive at the destination at a very slightly slower velocity because it still is subject, however minutely, to the gravitational field of the planet.
It will be traveling slower than the the unaffected object by exactly the same amount that it has to be so that they arrive at the same time since no additional energy was applied overall.
.

And what if I relocate point B far more away, so that planet is now closer to point A?
According to you the affected object will now lose the race?!

Does that mean that escape velocity is less than the entry velocity?

 

[rootone]

@phyzguy

See my second post in which I reconsidered this, we were probably posting at the same time.
It depends on at which point in the journey the tunnel planet exists.
If it exists exactly half way then it will at first accelerate the object, then decelerate it by the same amount.

@both
This is my reasoning, and please tell me where it is wrong:
The gravitationally affected object will be at first accelerated by the planet, and then will be decelerated by it as it moves on.
The affected object should arrive at a minutely slower velocity than the unaffected one
It is still declerating - that planet still is there tugging on it however minutely, although at an earlier stage in it's progress the object would have traveling at a higher velocity than the unnaffected object.
Since no overall energy is imparted to the gravitationally affected object as compared to the unaffected one, the net effect on time of arrival should be zero.

Location Location ...

 

[Bandersnatch]

@LM542, your intuition is correct (and rootone's answer wasn't). That you still seek confirmation shows that you need to see it in a mathematical form (so that there's no more doubt).

Do what berkeman advised you to do and draw a graph of V(t) of the two objects. Alternatively, write down the kinematic equation for velocity and calculate the delta V after two equal periods of acceleration with opposite signs. This will show you clearly whether the second object ever goes below the initial velocity.@rootone: the scenario asks for a situation where the acceleration is equal but opposite and with equal duration. As such, it is exactly equivalent to the runner example (only the curve of acceleration differs). In both cases the same amount of KE is added to the object and then removed.

If you were to engineer the situation so that the object begins its journey deeper in the gravity well than it finishes, the time of arrival could still be shorter than that for the unaccelerated object. What you need to do, is compare areas below the V(t) graph (i.e., take a definite integral). These represent the distance covered. You need equal areas (same distance) in the same interval for the object to arrive simultaneously.
A large positive spike can cover a lot more area than is later lost when the velocity goes below the initial value. Again, this is equivalent to the runner scenario, where one accelerates by a lot (gains head start), and then slows down to below the initial velocity.
Whether the other runner manages to catch up depends on exact setup of the situation.

But as was mentioned, the question in the OP is much simpler - the acceleration is symmetric.

 

[phyzguy]

Since no overall energy is imparted to the gravitationally affected object as compared to the unaffected one, the net effect on time of arrival should be zero.

No! No overall energy imparted means that both bodies have the same final velocity. But the time of arrival depends not on the final velocity but on the average velocity over the whole path. Since the gravitationally affected object always had a velocity greater than or equal to the unaffected object, its average velocity must be greater. So it will get there sooner. Suppose we both start our careers at the same time. You make $100,000/year for your whole career. I start out at $100,000/year, ramp up to $200,000/year, then ramp back down to $100,000/year. Who gets to a cumulative earnings of $1,000,000 first?

 

[rootone]

OK, my visualisation of what happens here is wrong.
I will try to explore the math of this a bit and try to find out why.

 

[LM542]

@LM542, your intuition is correct (and rootone's answer wasn't). That you still seek confirmation shows that you need to see it in a mathematical form (so that there's no more doubt).

Do what berkeman advised you to do and draw a graph of V(t) of the two objects. Alternatively, write down the kinematic equation for velocity and calculate the delta V after two equal periods of acceleration with opposite signs. This will show you clearly whether the second object ever goes below the initial velocity.

I have no clue about mathematics, so I can't confirm it.

Has this ever been observed in reality, could you please show me a proof or how this scenario is called, so that I can look it up.

 

[A.T.]

I have no clue about mathematics, so I can't confirm it.

The math here is very basic:

travel_duration = traveled_distance / average_velocity

So if average_velocity is more then travel_duration is less. Even if the initial and final velocities are the same.

But note that this applies only to slow objects. For example, for photons it would be the other way around, and the one in empty space would arrive first.

 

[Bandersnatch]

Has this ever been observed in reality

This is how all orbits work. It doesn't matter whether the orbiting object falls straight through the central body (it could be a tunnel, but it can also be just passing through due to the lack of interactions like dark matter is thought to do) or falls with some tangential velocity resulting in an elliptical orbit (more precisely: conic sections, so also parabolas and hyperbolas).
A body in a closed orbit returns to the point of origin with the same velocity as it started with. If it weren't so, and the velocity was actually lower than initial, the body would have too low a velocity to stay in the same orbit and would instead follow an inward spiral, and we'd have all planets plunge into the Sun.

This follows from the conservation of energy - an object in orbit around a massive body has gravitational potential energy associated with the distance from the central body and kinetic energy associated with its velocity. If it starts at the same distance and ends at the same distance, it must have also the same initial and final KE (so, same velocity).

As for the recommendations for further reading, look up SUVAT equations and energy conservation. These are taught at the secondary school level. There's plenty good tutorials on the net to guide you through them - I recommend Khan Academy. You could also try reading on orbital motion, but without a firm grounding in the more basic maths you won't be getting much out of it, understanding-wise.