Kylie a =E9crit :

Hi everyone,

How would I go about solving this problem?

A vehicle at stationary state (InitialVelocity =3D 0) starts moving from
point_1 toward point_2 (5 Km apart). It MUST stop completely (FinalVelo=

=3D 0) after travelling 5 km and reaching to point_2.

Maximum speed allowed is 25 km/h.

What would be required acceleration (to start moving from point_1) and
required deceleration (to stop moving at point_2) in order that vehicle
reach to point_2 in the shortest time.

Any help is much appreciated.

Max distance 5000 m
Max velocity 25 km/h =3D 6.95 m/s
Moving horizontally (orthogonal to gravitation)

You need to accelerate for the first half of the distance and
to decelerate for the second half

so, acceleration

a=3D 2v^2/d =3D ((6.95)^2)/2500 =3D 0.019321 m/s^2

Deceleration is the same.

Your vehicle will be there in the shortest time possible, given
a max velocity of 25 km/h.

Andr=E9 Michaud

Kylie wrote:

Hi everyone,

How would I go about solving this problem?


A vehicle at stationary state (InitialVelocity = 0) starts moving from
point_1 toward point_2 (5 Km apart). It MUST stop completely

= 0) after travelling 5 km and reaching to point_2.

Maximum speed allowed is 25 km/h.

There must also be a bound on acceleration and deceleration, since
otherwise the solution would be pretty obvious.

Scenario:

You accelerate at 1 g (9.8 m/s/s) to 25 km/h (6.94 m/s), maintain that,
and then decelerate at 1 g to rest, stopping exactly where required.
You maintain maximum allowable speed most of the way.

You accelerate at 10 g (98 m/s/s) to 25 km/h (6.94 m/s), maintain that,
and then decelerate at 10 g to rest, stopping exactly where required.
You maintain maximum allowable speed even longer.

If you could accelerate *instantaneously* (infinite m/s/s) to 25 km/h,
maintain that and stop instantly when you got to the other end (ouch!)
you would maintain *maximum speed* over the distance for maximum time,
and thus achieve minimum transit time of 12 minutes (= 5 km / 25 km/h =
0.2 hour).

See where this is going?

The required acceleration for minimum time is infinite, and would be so
*whatever* the maximum speed is.

Tom Davidson
Richmond, VA

tadchem wrote:

Scenario:

You accelerate at 1 g (9.8 m/s/s) to 25 km/h (6.94 m/s), maintain that,
and then decelerate at 1 g to rest, stopping exactly where required.
You maintain maximum allowable speed most of the way.

You accelerate at 10 g (98 m/s/s) to 25 km/h (6.94 m/s), maintain that,
and then decelerate at 10 g to rest, stopping exactly where required.
You maintain maximum allowable speed even longer.

If you could accelerate *instantaneously* (infinite m/s/s) to 25 km/h,
maintain that and stop instantly when you got to the other end (ouch!)
you would maintain *maximum speed* over the distance for maximum time,
and thus achieve minimum transit time of 12 minutes (= 5 km / 25 km/h =
0.2 hour).

See where this is going?

The required acceleration for minimum time is infinite, and would be so
*whatever* the maximum speed is.

Tom Davidson
Richmond, VA

Let it be known from the outset here: There is NO instantaneous
(infinite) acceleration. All motion is a change in position DURING
TIME.

If at least a couple of youse guys had paid attention to your algebra
lessons, you could have learned the equation for _average_
acceleration: a=vi-vt/(2t); not a=vi-vt/t; which is called
"instantaneous acceleration"; of which it is not.

Then you could have taught each other how to solve these sorts of
things.

Thanks everyone for wonderful responses!
Now I want to design a new problem:

This is actually a Robotic problem:

A robot at stationary state (InitialVelocity = 0) starts moving from
point_1 toward point_2.
It MUST reach to FinalVelocity of V at point_2.
We know ONLY the maximum velocity that robot can reach and the distance
between point_1 and point_2.
We have methods that can SET the following parameters of the robot:

AcceleratorPosition (i.e. Gas Pedal)
BrakePosition (i.e. Brake Pedal)

Of course the range of AcceleratorPosition and BrakePosition is known as
well.

At any time we can also GET the following information back from robot:

DistanceToPoint_2
Velocity
Acceleration

I was thinking of having a loop in the software, so that start increasing
AcceleratorPosition and every time GetVelocity until it reaches to the
maximum velocity and then stop increasing AcceleratorPosition so that
velocity remain constant.
In every run in the loop we also check to see what is the distance to
point_2. If this distance become less than "S", then
SettAcceleratorPosition to zero and start increasing BrakePosition until
it reaches to point_2 with the desired velocity of V.

Now the problem is how do I calculate "S"? Too many unknown! isn't it?

Any other idea?