Pytels Dynamics 12.8: Missile dynamics, acceleration and escape velocity

In summary, the missile will not return to the planet if its escape velocity is less than the minimum speed for which the gravitational force is equal to the pressure of air on the planet's surface.
  • #1
Alexanddros81
177
4

Homework Statement


A missile is launched from the surface of a planet with the speed v0 at t=0.
According to the theory of universal gravitation, the speed v of the missile after launch
is given by
v2 = 2gr0(r0 / r -1) + v02

where g is the gravitational acceleration on the surface of the planet and r0 is the mean
radius of the planet.
(a) Determine the acceleration of the missile in terms of r.
(b) Find the escape velocity that is the minimum value of v0 for which the missile will not return to the planet.
c)Using the result of part (b) calculate the escape velocity for earth, where g = 9.81m/s2 and
r0 = 6370km

Homework Equations

The Attempt at a Solution


I have tried part (a) . My attempt is found in the attached file.
Another way to go is to use the v2 = 2a(x-x0) + v02
and compare with given equation. I have tried that and couldn't get -g(r0/r)2
How do i solve part (b)? any hints
 

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  • #2
Alexanddros81 said:
I have tried part (a)
##\sqrt{a-b}## is not the same as ##\sqrt a- \sqrt b##.
Try differentiating without taking the square root.

For b, if it does not return, how large does r get?
 
  • #3
Check for part a and part b in attached files
If it doesn't return then r→∞
Is part b correct?
 

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  • #4
Alexanddros81 said:
Check for part a and part b in attached files
If it doesn't return then r→∞
Is part b correct?
Remarkably, you got the right answers to both a and b via flawed algebra.

In a, you differentiated the left hand side with respect to t and the right with respect to r. You must always do the same thing to both sides of an equation. If you differentiate both sides wrt t then on the right you get a factor ##\frac{dr}{dt}## (chain rule).
But in the next line, the derivative of v2 with respect to t is ##2v\frac{dv}{dt}=2va##, not just 2a. These two errors canceled out to give the right result.

In b, you again have made the blunder of distributing √ across a sum of terms. ##\sqrt{v^2+2gr_0}## is not ##\sqrt{v^2}+\sqrt{2gr_0}##. Is √1+√4=√(1+4)? You got away with that because in the case you are interested in v=0.
 
  • #5
Ok got it!
On the left side we let z=v2 and take the dz/dt = (dz/dr)(dr/dt) = 2va
On the right side we do the same : 2gr02 dz/dt where z=1/r => 2gr02(dz/dr)(dr/dt) =
= 2gr02(-1)r-2v
the 2 and v cancel out
This is implicit differentiation isn't it?
 
  • #6
Alexanddros81 said:
Ok got it!
On the left side we let z=v2 and take the dz/dt = (dz/dr)(dr/dt) = 2va
On the right side we do the same : 2gr02 dz/dt where z=1/r => 2gr02(dz/dr)(dr/dt) =
= 2gr02(-1)r-2v
the 2 and v cancel out
This is implicit differentiation isn't it?
Yes. But I think you mean dz/dt=dz/dv dv/dt on the left.
 
  • #7
yes.
 

1. What is Pytels Dynamics 12.8 and how does it relate to missile dynamics?

Pytels Dynamics 12.8 is a mathematical model used to analyze the dynamics of missiles, specifically their acceleration and escape velocity. It takes into account factors such as the missile's mass, thrust, and aerodynamics to predict its motion and behavior.

2. What is acceleration and why is it important in missile dynamics?

Acceleration is the rate of change of an object's velocity over time. In missile dynamics, it is important because it determines how quickly a missile can change its speed and direction, which ultimately affects its trajectory and target accuracy.

3. How is escape velocity calculated using Pytels Dynamics 12.8?

Escape velocity is the minimum speed required for an object to escape the gravitational pull of a larger body, such as a planet. In Pytels Dynamics 12.8, it is calculated by considering the gravitational force of the planet, the mass of the missile, and the atmospheric drag on the missile.

4. What are the main limitations of Pytels Dynamics 12.8 in modeling missile dynamics?

Like any mathematical model, Pytels Dynamics 12.8 has certain limitations in accurately predicting missile dynamics. Some of these limitations include not accounting for external factors such as wind, turbulence, and air density variations, and not considering the effects of real-life obstacles and terrain on the missile's trajectory.

5. How can Pytels Dynamics 12.8 be applied in real-world scenarios?

Pytels Dynamics 12.8 can be used in various real-world scenarios, such as designing and testing missiles, analyzing the performance of existing missile systems, and predicting the behavior of missiles in different atmospheric and gravitational conditions. It can also be applied in the field of aerospace engineering to study the dynamics of spacecraft and other flying vehicles.

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