Solving DE Problems: Balloon, Projectile & Trypsin Formation

  • Thread starter uknowwho
  • Start date
In summary: Have you covered the topic of finding extrema of a function in calculus? If so, you should know how to do this.
  • #1
uknowwho
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1. Homework Statement

. A balloon is rising at the constant rate of 10 feet/second and is 100 feet from the ground at the instant when the astronaut drops his binoculars. (a) How long will it take the binoculars to strike the ground? (b) With what speed will the binoculars strike the ground?


2. A projectile is fired vertically upward by a cannon with an initial velocity of vo meters per second.At what speed will the projectile be moving when it returns and strikes the hapless cannoneer(Neglect air resistance)


3. Consider the differential equation dy/dt =k(A-y)(B+y) for the formation of trypsin in the small intestine.Assuming that A>B determing the time t at which trypsin is being formed most rapidly.




Homework Equations





The Attempt at a Solution



for (1)
a=-g
v=-gt + c1

v=0 and t=0

0=0 +c1
so c1=0

v=-gt

s=-gt^2/2 +c2

s=100 t=0

so 100=c2

s=-gt^2/2 + 100
s=-4.9t^2 + 100

putting s=0

-4.9t^2+100=0

getting t=4.5sec

and v=-44feet/sec

but the answer given at the back is t=2.83sec and v=80.62ft/sec

what am i doing wrong
When binoculars will be dropped at that time v will be zero and so will be the time and at the same time its distance from the ground will be 100feet
vi=v0 m/sec

for (2)

acc=-g
v=-gt+c1
v0=c1
v=-gt+v0

s=-gt^2/2 + v0t + c2

s=0 and t=0

c2=0

s=-gt^2/2 +vot

s=-4.9t^2 + vot

how to solve it further?

for (3)

dy/dt=k(A-y)(B+y)

dy/dt+k(AB+Ay-By-y^2)

dy/dt=ABk +Aky_Bky-ky^2
dy/dt=ABk + k(A-B)y-ky^2

dy/k(A-B)y-ky^2=(ABk) dt

Im stuck here
 
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  • #2
An astronaut in a balloon in a height of 100ft? Well, whatever.
The initial velocity of the binoculars is not 0 as the balloon is rising, and you are mixing feet and meters (in g) here. This would be obvious if you would work with units, which is a good idea in physics.

(2) is easy if you consider the symmetry of the setup, or use energy conservation. If you want to go the long way, set s=0 and solve for t. v follows as a result of t.

In (3), it is impossible to calculate t without initial conditions. It is possible to calculate y where the change of y reaches its maximum.
 
  • #3
mfb said:
An astronaut in a balloon in a height of 100ft? Well, whatever.
The initial velocity of the binoculars is not 0 as the balloon is rising, and you are mixing feet and meters (in g) here. This would be obvious if you would work with units, which is a good idea in physics.

(2) is easy if you consider the symmetry of the setup, or use energy conservation. If you want to go the long way, set s=0 and solve for t. v follows as a result of t.

In (3), it is impossible to calculate t without initial conditions. It is possible to calculate y where the change of y reaches its maximum.

for (1) if i put v=10,t=0 and s=100,t=0
i get v=-gt+10

and s=-4.9t^2+10t+100

-4.9t^2+10t+100=0

i get t as 5.65s

these should be the intial cdonditions?

for (3) how to calculate y where the change of y is max?
can you be more precise
 
  • #4
I agree with your formulas for (1), but you are still mixing feet and meters.

for (3) how to calculate y where the change of y is max?
How can you calculate the point where something is maximal in general? Just use that.
 
  • #5
Do you know what g is in units of ft/s^2?
 
  • #6
mfb said:
I agree with your formulas for (1), but you are still mixing feet and meters.

How can you calculate the point where something is maximal in general? Just use that.


I solved the other two but (3) I'm still not able to do..Can you just show the starting steps to give me an idea? it would be better if you could solve it so that I can understand
 
  • #7
I am sure you calculated the maximum of functions before. How did you do it? Just do the same here.

I don't think it would be useful if I solve it. It is against the forum rules, and you can find many solved examples of similar problems anyway.
Did you have a look at the derivative?
 
  • #8
uknowwho said:
for (3) how to calculate y where the change of y is max?
So you're saying you want to find where the function dy/dt attains a maximum. As mfb has suggested, that should ring some bells.
 

FAQ: Solving DE Problems: Balloon, Projectile & Trypsin Formation

1. How do I solve a DE problem involving a balloon?

To solve a DE problem involving a balloon, you will first need to understand the forces acting on the balloon, such as buoyancy and air resistance. Then, you can use the ideal gas law and the differential equation for the rate of change of the balloon's volume to solve for the balloon's position and velocity over time.

2. What is the difference between projectile motion and trypsin formation?

Projectile motion involves the motion of an object through the air, while trypsin formation refers to the process of creating trypsin, a protease enzyme, through the reaction of trypsinogen with an activating enzyme. Both can be modeled using differential equations, but they involve different physical phenomena.

3. What are the key steps in solving a DE problem?

The key steps in solving a DE problem are: identifying the variables and parameters involved, writing a differential equation that describes the relationship between these variables, determining initial conditions, and using mathematical techniques such as separation of variables or substitution to solve the DE and find a general solution. You may also need to apply boundary conditions to find a particular solution.

4. How do I know if my solution to a DE problem is correct?

To check if your solution to a DE problem is correct, you can plug the solution back into the original differential equation and see if it satisfies the equation. You can also compare your solution to known solutions or use numerical methods to approximate the solution and compare it to your result.

5. Can I use a computer to solve DE problems?

Yes, you can use a computer to solve DE problems. There are various software programs and programming languages that can be used to numerically solve DEs, such as MATLAB, Mathematica, and Python. These tools can also graph the solutions and provide visualizations of the DE problem.

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