Line Integrals for trajectories

In summary, the conversation discusses the concept of finding the work done by a vector field on a particle moving along a path using a line integral. The question is raised if this applies to projectiles and for a real-life example, it is suggested that wind could be considered. However, it is noted that the path of wind would constantly change. It is concluded that any field will affect the path and a constrained path, such as a bead on a wire, would require the consideration of the force exerted by the wire.
  • #1
saminator910
96
1
So I was wondering if I defined a vector field F, and a Trajectory of a particle x=t y=.5at^2+vit+si
and I can find the work done by the field on a particle moving on a path with a line integral ∫F.dr, so what would this equate to for a projectile does it apply to this?, could you give me a real life example, could it possibly be wind? but wind would change it's path right? Thank you for any response.
 
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  • #2
Any field will affect the path. You could consider a particle on a constrained path, like a bead on a wire, but then you need to bring in the force exerted by the wire.
 

1. What is a line integral for trajectories?

A line integral for trajectories is a mathematical concept used to calculate the amount of a specific quantity, such as force or work, along a given path or trajectory. It takes into account both the magnitude and direction of the quantity being integrated.

2. How is a line integral for trajectories different from a regular integral?

A line integral for trajectories differs from a regular integral in that it is calculated along a specific path or trajectory, rather than over a continuous area or volume. It also takes into account the direction of the quantity being integrated, while regular integrals only consider magnitude.

3. What is the significance of line integrals for trajectories in physics?

Line integrals for trajectories are important in physics because they allow us to calculate the work done by a force along a specific path, rather than just the overall work done. This is particularly useful in situations where the force may vary along the path, as it allows us to take into account the changing direction and magnitude of the force.

4. How do you calculate a line integral for trajectories?

To calculate a line integral for trajectories, you first need to parameterize the path or trajectory in terms of a single variable. Then, you integrate the function to be integrated with respect to this variable, and evaluate the integral at the limits of the path. This can be done using various techniques, such as the fundamental theorem of calculus or Green's theorem.

5. In what fields of science are line integrals for trajectories commonly used?

Line integrals for trajectories are commonly used in fields such as physics, engineering, and mathematics. They are particularly useful in analyzing the motion of particles and understanding the work done by forces in a given system. They also have applications in fluid dynamics, electromagnetism, and other areas of physics.

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