Calculate Constant Velocity & Speed for Trapezoid Move

[robertor]

Hi All,

A question to calculate the speed at constant velocity during a Trapezoid move.

An object moves from standstill at a known acceleration (1m/s/s) to a constant velocity (unknown) for a period (unknown) and then decelerates at a known deceleration (2m/s/s) to standstill. The total displacement is known (10m) and the total time is known (25s). How can I calculate the constant velocity and therefore the speed at constant velocity during the trapezoid move?

Thanks!
Rob

 

[tiny-tim]

welcome to pf!

Hi Rob! Welcome to PF!

Call the times t₁ t₂ and t₃

then write out all the equations …

show us what you get. ​

 

[robertor]

Which equation?

Thank you for the quick reply Tim!

I don't really know where to start due to the amount of unknowns...

If I begin with finding x for t₁:

x = v0t₁ + ½at₁²
x = 0*t₁ + ½*1*t₁²
x = ½*t₁²

Is this in the right direction for t₁?

 

[tiny-tim]

yes

now find the speed at t₁, which will be the starting speed for the next part,

and then write the equation for t₂

 

[robertor]

Constant Acceleration - Final Velocity

Final Velocity
v₂ = v₁ + at
v₂ = at

This gives me final velocity after the constant acceleration...but I am at a bit of a loss as to what to do here. Thanks for the help so far, you are very good!

 

[robertor]

Constant Velocity Displacement

Does this mean that the displacement during the constant velocity is calculated:
x₂ = (at)²

 

[tiny-tim]

(been out all evening! )

Final Velocity
v₂ = v₁ + at
v₂ = at

This gives me final velocity after the constant acceleration...but I am at a bit of a loss as to what to do here. Thanks for the help so far, you are very good!

(what is  ?)

yes v₁ = v₂ = t₁

t₂ is unknown

now work backwards from the end, to find the speed at the start of the deceleration, the same way you worked forwards from the start to find the speed at the end of the acceleration

 

[robertor]

Decel Part

OK, so working backwards for t3:

x = v0t₃ + ½at₃²
x = 0*t₃ + ½*2*t₃²
x = t₃²

The equation is simplified to this only because of the value that I passed in, had the decel not been 2m/s/s, the equation would look different, obviously!

 

[tiny-tim]

ok, now add all the x's, and all the t's

 

[robertor]

OK, so adding...

ACCEL: x = ½*t₁²
C.VEL: x = (at₂)²
DECEL: x = t₃²

x = ½*t₁² + (at₂)² + t₃²

 

[tiny-tim]

(just got up :zzz:)

C.VEL: x = (at₂)²

no, x = vt₂

(and then put the total displacement = 10, and write out total time = 20)

 

[robertor]

Simplifying

Thank you for the info so far!

x = (½*t₁²) + (vt₂) + (t₃²)
10 = (½*t₁²) + (vt₂) + (t₃²)

But I only have total time, and each of these time periods are separate, so how would I "put it" total time=20 into this equation?

 

[tiny-tim]

… how would I "put it" total time=20 into this equation?

you have to learn to translate from english to maths!

"total time" -> t₁+ t₂ + t₃ !​

(and now you'll need those two equations for v)

 

[robertor]

Sure, well I know:

t = t₁ + t₂ + t₃
x = 10

So,

10 = (½*t₁²) + (vt₂) + (t₃²)

But I am stuck here. If during static velocity (t₂) v=x₂/t₂ then v = tx₂. But this doesn't really help does it?

 

[tiny-tim]

you have t₁+ t₂ + t₃ = 25

and (½*t₁²) + (vt₂) + (t₃²) = 10

but you don't know what v is, sooo …

now you use your two equations for v, the one working forwards from the start, and the other working backwards from the finish! ​

(i'm going out now for the rest of the day)

 

[robertor]

Confused...

But I only need to find v, which is not part of the acceleration period or deceleration period, so why do the first two equations have anything to do with v (the constant velocity period)?

 

[tiny-tim]

you need v for the total displacement equation,

and you find v (as a function of t₁ or t₃) from the acceleration and deceleration equations

 

[robertor]

This equation?
v=v₀+at

 

[tiny-tim]

yes, use it forwards from the start, and backwards from the finish

 

[robertor]

Im really stuck...
I can't seem to figure out how v=v₀+at for the acceleration deceleration period will help me find v? Am i being stupid? I have been looking at this for hours and can't seem to slot the final pieces together.

t₁+ t₂ + t₃ = 25
(½*t₁²) + (vt₂) + (t₃²) = 10

How does using v=v₀+at help me from here?

 

[tiny-tim]

How does using v=v₀+at help me from here?

let's take the first stage …

that's v = 0 + 1*t₁, so you can substitute that into

(½*t₁²) + (vt₂) + (t₃²) = 10​

which was a four-unknowns equation (t₁ t₂ t₃ and v)

but now becomes a three-unknowns equation (t₁ t₂ and t₃ )

then you use v = v₀ + at again, for the last stage, and that gives you an equation relating v and t₃, so you can eliminate one more unkown

 

[robertor]

I see..

Oh I see, sorry i was confused as v was the constant velocity during t2, and I was substituting in the velocity as a result of acceleration, but you made me realize that this velocity is the final velocity of the acceleration period

v=v₀+at₁
v=at₁

So inserting this into the equation:

(½*t₁²) + (a*t₁*t₂) + (t₃²) = 10

For the decel:

v=v₀+at₃
v=2*t₃
t₃ = v/2
t₃ = (at₁)/2

So putting this into the equation:

(½*t₁²) + (a*t₁*t₂) + (((at₁)/2)²) = 10

(½*t₁²) + (a*t₁*t₂) + ((at₁)/4) = 10

 

[tiny-tim]

(½*t₁²) + (a*t₁*t₂) + (((at₁)/2)²) = 10

(½*t₁²) + (a*t₁*t₂) + ((at₁)/4) = 10

yes , except:

i] i think you meant ((a²t₁²)/4)

ii] you know that a = 1​

now simplify that other equation, t₁ + t₂ + t₃ = 25

 

[robertor]

Apologies, I typed it incorrectly.

(1/2*t₁²) + (1*t₁*t₂) + ((1²t₁²)/4) = 10

(1/2*t₁²) + (t₁t₂) + ((2t₁²)/4) = 10

(1/2*t₁²) + (t₁t₂) + ((t₁²)/2) = 10

(t₁²) + (t₁t₂) = 10

I still have 2 unknowns, but I also know t₁ + t₂ + t₃ = 25. I have got to a stand still again!

 

[tiny-tim]

hi robertor!

I still have 2 unknowns, but I also know t₁ + t₂ + t₃ = 25.

ok, now eliminate t₃ from that equation

 

[robertor]

how?

 

[tiny-tim]

same way you eliminated it from (½*t₁²) + (a²*t₁²*t₂) + (t₃²) = 10

(you are allowed to use things twice! )

 

[robertor]

Sorry, I read it over and I realize how dumb I sounded! I used this only a few steps ago!

So I have:
t₃ = (at₁)/2
t₃ = (2*t₁)/2
t₃ = t₁

t₁ + t₂ + t₃ = 25
t₁ + t₂ + t₁ = 25
(2t₁) + t₂ = 25

(t₁²) + (t₁t₂) = 10

 

[tiny-tim]

I used this only a few steps ago!

he he!

(2t₁) + t₂ = 25

(t₁²) + (t₁t₂) = 10

right, so t₂ = 25 - 2t₁,

and you can substitute that into the second equation

 

[robertor]

Simplifying Down...

(t₁²) + (t₁(25 - 2t₁) = 10

 

[tiny-tim]

ok, that's a quadratic equation in t₁

 

[robertor]

(t₁²) + (t₁(25 - 2t₁) = 10
t₁² + (25*t₁ - 2t₁²) = 10
25*t₁ - 1t₁² = 10
25*t₁ - 1t₁² - 10 = 0
-25*t₁ + t₁² + 10 = 0
-25*t₁ + t₁² = -10
-25*t₁ + t₁² = -10
-25*t₁ + t₁² + 156.25 = -10 + 156.25
-25*t₁ + t₁² + 156.25 = 146.25
(t₁ + -12.5)(t₁ + -12.5) = 146.25
t₁ + -12.5 = 12.093386622 OR -12.093386622
t₁ = 24.593386622
t₁ = 0.406613378

 

[tiny-tim]

t₁ = 24.593386622
t₁ = 0.406613378

yes, that looks ok: you have two possible solutions for t₁, so now find t₂ and t₃ in each case and check that it all works

(for something this long, there's very likely a mistake somewhere! )

finally, the original question was …

A question to calculate the speed at constant velocity during a Trapezoid move.

… so find v

btw …

(t₁²) + (t₁(25 - 2t₁) = 10
t₁² + (25*t₁ - 2t₁²) = 10
25*t₁ - 1t₁² = 10
25*t₁ - 1t₁² - 10 = 0
-25*t₁ + t₁² + 10 = 0
-25*t₁ + t₁² = -10
-25*t₁ + t₁² = -10
-25*t₁ + t₁² + 156.25 = -10 + 156.25
-25*t₁ + t₁² + 156.25 = 146.25
(t₁ + -12.5)(t₁ + -12.5) = 146.25
t₁ + -12.5 = 12.093386622 OR -12.093386622

most people would just have written

(t₁²) + (t₁(25 - 2t₁)) = 10
t₁² - 25*t₁ + 10 = 0
so t₁ = 12.5 ± √(156.25 - 10)
= 24.59 or 0.41 …​

… is there any reason why you've avoided that?

 

[robertor]

Finally...

Solution A:
t₁ = 24.593386622
v = v₀ + at
v = 1 * 24.593386622
v = 24.593386622m/s

Solution B:
t₁ = 0.406613378
v = v₀ + at
v = 1 * 0.406613378
v = 0.406613378m/s

… is there any reason why you've avoided that?

I found it easier to do the steps logically:
t1*25 - 2t₁²

And then work out the steps one by one, moving the constant to the left hand side, so the result is 0. Multiplying by -1 to remove the negative figures. Half of part a, squared added to both sides. Thats the way I was taught in I think? Your method seems more concise however I fail to see the middle steps you took, possible because I am nowhere near as advanced as you are!

 

[tiny-tim]

Solution A:
t₁ = 24.593386622
v = v₀ + at
v = 1 * 24.593386622
v = 24.593386622m/s

Solution B:
t₁ = 0.406613378
v = v₀ + at
v = 1 * 0.406613378
v = 0.406613378m/s

yes that looks fine …

now round it off to a sensible number of significant figures, and you're finished!

(and, as i said before, there's probably a mistake somewhere, so you'd better check it very carefully)

Thats the way I was taught in I think? Your method seems more concise however I fail to see the middle steps you took …

i used the standard quadratic equation formula …

first convert the equation to ax² + bx + c = 0, and then the solution is x = (-b ± √(b² - 4ac))/2 …

it's completely acceptable in exams, so you should learn it! ​

ok, lets' go over how you did it, so you know what to do in future …

first, you translated some of the question from maths into english … that gave you two equations

second, you used physics to get two more equations, making 4 equations with 4 unknowns

third, you started eliminating one unknown at a time until you got to 1 equation with only 1 unknown

 

[robertor]

Rounding

v = 24.59m/s OR v = 0.41m/s

I mean looking at it, I know which is more likely, is there a way to confirm which of these two values is correct?

 

[tiny-tim]

… looking at it, I know which is more likely, is there a way to confirm which of these two values is correct?

yup!

something comes out negative when it can't be

in most problems, you'd find that t₁ has one positive solution and one negative solution …

obviously, you reject the negative solution!​

but in this problem, it's slightly different …

can you see what it is? ​

 

[robertor]

I could calculate the displacement during t1, t2 and t3. They should add up to my known displacement, if they don't, then its the other one?

 

[tiny-tim]

no

what is the value of t₂ in each case? ​

 

[robertor]

Sure I can calculate the value of t₂ in this scenario, however my next task is to rewrite this equation as a generic formula to find t₁, using the same technique you have shown me. Therefore, sure I can look at t₂ in both cases but how will I know which one is correct. If I provide accel, decel, total time and total displacement, what happens if the object does not have time to accelerate to a constant velocity, and the shape within the velocity x time graph becomes a triangle? Surely t₂ could be negative using this current equation. There must be a more generic method for checking which value of V is correct, which is why I suggested adding up the displacements or the times to see if they equal the displacement or time provided at the beginning :)

 

[tiny-tim]

… which is why I suggested adding up the displacements or the times to see if they equal the displacement or time provided at the beginning :)

that can't possibly work: the correct values for total time and displacement were part of your input

Therefore, sure I can look at t₂ in both cases but how will I know which one is correct.

calculate both values, and then it should be obvious

what are they?​

 

[robertor]

Sure on this occasion the correct answer is obvious. However what I was pointing out is that the answer will not always be obvious, am I wrong?

 

[tiny-tim]

Sure on this occasion the correct answer is obvious. However what I was pointing out is that the answer will not always be obvious, am I wrong?

i'm not entirely convinced that you have got the correct answer, since you haven't specifically said what it is

the general rule is that if there's two solutions, then they're both valid …

why would they not be??​

the only exception is where something is undefined for particular values (in this case for negative values) of some variable

 

[robertor]

t₂

t₂ = 24.390079932 OR t₂ = -11.890079933

 

[tiny-tim]

yup! and, as i think you've noticed, t₂ can't be negative (unless you redefine the question considerably), so the second solution has to be rejected

(but plenty of problems do have two solutions, eg if you throw the ball at 5 m/s, at what angle should you throw it to get it through the hoop?)

 

[robertor]

What if both answers are positive?

 

[tiny-tim]

then both answers are valid