Trajectory of objects in imperfect conditions.

In summary, the conversation discusses the challenges of modeling the motion of a projectile launched at a high altitude where the curvature of the Earth and changing gravity play a role. It is a field of study called exterior ballistics and involves solving complicated differential equations. The accuracy of the Paris Gun, a projectile launched during WWI, was affected by various factors such as changes in gravity, density of the atmosphere, and the condition of the propellant. Ultimately, the Paris Gun had little strategic value and was mainly used for psychological purposes.
  • #1
saminator910
96
1
I am familiar with standard distance-time models for paths of projectiles in perfect conditions, ie, where the curvature doesn't play a role, and where gravity is constant no matter the height. My question is what if you launch a projectile so high that the curvature of Earth plays a role, and gravity varies as you increase and decrease height, is there and way to model it's motion, say a distance to surface-time equation? It would probably be similar to basic ones like s(t)=.5at^2+v*t, but since the acceleration changes over time it becomes difficult. It seems the acceleration at any point would be (Fg-Fc)/m (force of gravity minus force centripital, divided my mass), but then you get distance in the equation twice... Also, since (mv^2)/r = Fc how would you know the tangential velocity? Do equations already exist for this?
 
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  • #2
Basically, at this point, you are dealing with orbital motion. Kepler's Laws will give you the basic trajectory. From there, you'll have to look into more detailed orbital mechanics for things like time of flight, velocities at each point, and so on.

Of course, in real world, if velocity is high enough for these things to matter, your bigger concern will be drag. It is difficult enough to account for drag with level ground and constant gravity. There are methods, though. Orbital mechanics with drag pretty much have to be solved with numerical methods. If gravity changing with height is a factor, density changing with altitude will certainly be an even bigger factor. That means having a pretty good barometric model on top of which you'll be running your simulation. It gets tricky.
 
  • #3
It's a field of study called exterior ballistics, and the work usually involves the numerical solution of complicated differential equations of motion.

In WWI when the Germans were firing their Paris gun, they had to account for, among other variables, changes in gravity, density of the atmosphere, curvature of the earth, coriolis forces, the gravitational effect of the moon, the temperature, and the condition of the propellant.
 
  • #4
SteamKing said:
In WWI when the Germans were firing their Paris gun, they had to account for, among other variables, changes in gravity, density of the atmosphere, curvature of the earth, coriolis forces, the gravitational effect of the moon, the temperature, and the condition of the propellant.

It's my understanding that the accuracy of the Paris Gun was terrible. The few projectiles that were fired landed somewhere in a 10 kilometer radius around the center of Paris (so they still landed in the suburbs).

For calculating the amount of propellent needed the crudest approximation would have been sufficient. Density of the atmosphere at different stages of the flight: probably yes, the projectiles climbed tens of kilometers high. My understanding is that all other effects were totally swamped.

The gun could fire 10 projectiles or so, then it had to be shipped back to the factory to resurface the inside of the barrel.
In terms of strategic value the Paris Gun was a waste of resources. The effect was psychological: the fact that the Germans were able to reach Paris with that Gun.
 
  • #5


Thank you for your question. The trajectory of objects in imperfect conditions, such as when the curvature of the Earth and varying gravity are involved, can be quite complex and difficult to accurately model. This is because the forces acting on the object are constantly changing and the equations used for perfect conditions may not apply.

There are indeed equations that exist for these types of scenarios, known as orbital mechanics equations. These take into account the varying gravity and curvature of the Earth to accurately predict the trajectory of an object. However, these equations can be quite complex and require advanced mathematical knowledge to understand and use.

In general, the trajectory of an object in such conditions can be modeled as a combination of its tangential velocity and centripetal acceleration. The tangential velocity is the velocity at which the object is moving along its curved path, while the centripetal acceleration is the acceleration towards the center of the curvature.

To find the tangential velocity, you can use the equation (mv^2)/r = Fc, where m is the mass of the object, v is the tangential velocity, r is the radius of the curved path, and Fc is the centripetal force. This equation can be rearranged to solve for v.

However, as you mentioned, the acceleration at any point would be (Fg-Fc)/m, where Fg is the force of gravity and Fc is the centripetal force. This means that the acceleration is constantly changing and can make it difficult to model the trajectory accurately.

In order to accurately model the trajectory, it is important to take into account all the forces acting on the object, including air resistance and any external forces. It is also important to use numerical methods and computer simulations to accurately predict the trajectory.

In summary, while equations do exist for modeling the trajectory of objects in imperfect conditions, it can be quite complex and difficult to accurately predict due to the constantly changing forces. It is important to take into account all the factors and use advanced mathematical methods to accurately model the trajectory.
 

1. What factors affect the trajectory of objects in imperfect conditions?

The trajectory of objects in imperfect conditions can be affected by a variety of factors such as air resistance, wind speed and direction, surface friction, and the shape and weight of the object.

2. How does air resistance impact the trajectory of objects?

Air resistance, also known as drag, can cause objects to slow down and change direction as they move through the air. It can also cause objects to spin or tumble, making it difficult to predict their trajectory.

3. Can the shape of an object affect its trajectory in imperfect conditions?

Yes, the shape of an object can greatly impact its trajectory. Objects with streamlined shapes, such as airplanes or rockets, are designed to minimize air resistance and maintain a stable trajectory. On the other hand, irregularly shaped objects may experience more air resistance and have a less predictable trajectory.

4. How does surface friction affect the trajectory of objects?

Surface friction, or the resistance an object encounters when moving along a surface, can slow down or alter the trajectory of an object. For example, a ball rolling on a rough surface will experience more friction and may not roll as far as it would on a smooth surface.

5. How can we calculate the trajectory of an object in imperfect conditions?

To calculate the trajectory of an object in imperfect conditions, we must take into account the various factors that can impact its path, such as air resistance, wind, and surface friction. This can be done using mathematical equations and formulas, or through computer simulations.

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