# Distance vs displacement graph

1. Jun 9, 2015

### takando12

1. The problem statement, all variables and given/known data
The V-S graph ( x axis as S and y axis as V)of a airplane on a runway is as follows: for 0-100 meters the velocity increases steadily in a straight line from 0-40 m/s. For the remaining 100 -200 meters, the velocity increases steadily from 40 -50m/s in a straight line. The slope is more in the first half. I need to find the acceleration of the plane at S=50m and S=150 m. I also need to draw an A-S graph.

2. Relevant equations

3. The attempt at a solution
I tried to find out what the slope dv/ds was . Using the chain rule. dv/dt * dt/ds= a/v and that's 1/t. Didn't do anything else productive. Any help?

2. Jun 9, 2015

### Alettix

Hi!

I think using the chain rule is a good start, but I would do it this way: dv/ds * ds/dt = dv/dt
Now, from the graph we know that dv/ds = K (a constant).
Rewriting dv/dt = d^2s/dt^2, this should give us a solvable differential equation I think. :)

3. Jun 9, 2015

### Orodruin

Staff Emeritus
This assumes constant acceleration from $v = 0$ at $t = 0$, which is not the case in the second part. However, your start is good. You now have
$$\frac{dv}{ds} = \frac av.$$
From your information, you should be able to find both $dv/ds$ and $v$ at any given point and from there be able to solve for $a$.

4. Jun 9, 2015

### DEvens

Hmmm.... I am getting an interesting answer when I try to work out when the plane started at S=0.

5. Jun 9, 2015

### takando12

Can we split in into two parts with two different constant accelerations and solve it?
1) At 50 m (0-100 interval)
assuming const. acceleration a , v2-u2=2as
a = 1600/100*2
a = 8m/s2
2) 100m (100-200 interval)
const.acceleration a1 v2-u2=2a1s
2500-1600= 2*100*a1
a1=4.5m/s2
this matches the answers in the back of my textbook( ecstasy...) but i don't understand exactly. Why is the acceleration constant (in two separate parts) because dv/ds=1/t? please bear with me, I am completely new to calculus and using it in physics.

6. Jun 9, 2015

### takando12

sorry, i'm kinda new to calculus. I get everything you said , but I don't know how to proceed from d^2s/dt^2. A little help please?

7. Jun 9, 2015

### DEvens

No, the acceleration is not constant.

I will give you a hint. What is the derivative with respect to $t$ of $e^{At}$ ? So the problem has (for the first part) $V = A S$.

8. Jun 9, 2015

### Orodruin

Staff Emeritus
No, the acceleration is not constant in either part. You are overcomplicating things. You already have v as a function of s as well as an expression for the acceleration which only involves this function. There is no need to involve time or any other assumptions.

9. Jun 9, 2015

### Orodruin

Staff Emeritus
In my opinion, involving the time in the solution (other than in applying the chain rule as above) is only an unneseccary complication. There is no need to find the displacement or velocity as a function of time.

10. Jun 9, 2015

### takando12

I really do not understand what to do after that. at s=50m , v=20m ? then?
But why is the answer turning out to be right with my previous post ? i checked the back again . Is it just fluke?

11. Jun 9, 2015

### Orodruin

Staff Emeritus
Yes, it is a fluke that results from the point being chosen in the middle of the interval. The correct approach should give you the same end result. A pretty non-pedagogical task, which gives the same results with the wrong approach ...

12. Jun 9, 2015

### takando12

Oh. It turns out to be the average acceleration which is in the middle?
Coming to the right method. I need help please, i don't know how to proceed after dv/ds= a/v. At s= 50 m , I suppose v= 20m/s. what then?

13. Jun 9, 2015

### Orodruin

Staff Emeritus
Then you need to compute dv/ds. This should be easy, since v is a linear function of s.

14. Jun 9, 2015

### takando12

okay, dv/ds for the first half is 40/100=2/5
a/v=2/5 and v=20.Hence v=8m/s2. That right?
Just a few things cleared up:
1) Does the fluke work because it's average acceleration?
2) And i need to draw the a-s graph as well. Must i just use the values of a and s that i computed and draw it ?Or is there a more formal method involving calculus?

15. Jun 10, 2015

### Orodruin

Staff Emeritus
Well, you have already concluded that dv/ds is constant over each part and you know that v is a linear function of s. What does this mean for a?

16. Jun 10, 2015

### takando12

Well....is the acceleration is increasing in the first half as the a/v slope is constant and the velocity is increasing ?and then the a/v slope gets smaller so, the acceleration decreases?

17. Jun 10, 2015

### Orodruin

Staff Emeritus
The easiest way to understand it is to just multiply both sides by v to obtain $a = v (dv/ds)$. In both segments, $dv/ds$ is constant and $v$ is a linear function. You are correct that the acceleration decreases in the transition between the two sections, but then it increases again as $dv/ds$ is again constant.

18. Jun 10, 2015

### takando12

Sorry for the late reply. Yes it makes sense for it to increase again. Well that's about it........thank you for all the help and for bearing with my infinite stupidity!!
Merci monsieur Orodruin. Good day.

19. Jun 10, 2015

### Orodruin

Staff Emeritus
There is nothing stupid with spending effort on trying to understand something.

20. Jun 11, 2015

### takando12

Well said sir.