Calculating Projectile Trajectories with Gravitational and Electrostatic Forces

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Discussion Overview

The discussion revolves around calculating the trajectories of two positively charged projectiles influenced by both gravitational and electrostatic forces. Participants explore the complexities of projectile motion under these forces, including the effects of variable acceleration due to electrostatic repulsion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • An undergraduate physics student presents a scenario involving two charged projectiles, detailing their mass, charge, launch angle, and initial velocity.
  • One participant notes that the gravitational force acts vertically while the electrostatic force acts horizontally, suggesting that the two forces can be treated independently in the analysis.
  • Another participant points out that the electrostatic force is inversely proportional to the square of the separation distance, raising the question of how to solve the resulting differential equation.
  • A participant expresses confusion regarding the non-constant nature of acceleration and separation, indicating a challenge in the calculations.
  • There is a suggestion to derive an energy equation to facilitate solving the problem, although one participant acknowledges the complexity of this approach.

Areas of Agreement / Disagreement

Participants generally agree on the independence of gravitational and electrostatic forces in the analysis, but there is no consensus on the correct method to solve the equations governing the projectile motion, with differing opinions on the validity of proposed solutions.

Contextual Notes

The discussion highlights the challenges of dealing with variable forces and the complexities of solving differential equations in this context. Specific assumptions about the system, such as ignoring air resistance, are implied but not explicitly stated.

Who May Find This Useful

This discussion may be useful for undergraduate physics students, educators, and anyone interested in the dynamics of charged particles and projectile motion under the influence of multiple forces.

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I am an undergraduate physics student and found myself thinking about a scenario where two positively charged projectiles are fired at an angle to the horizontal at a velocity with a certain separation. So they will be subject to a gravitational force and perpendicular to this, an electrostatic force.
The distance dependent acceleration is something I have not covered before and am unsure as to how to calculate where the projectiles will land.

Lets give the projectiles a mass of 1kg and a charge of +1mC. Say they are fired at 45degrees to the horizontal with a velocity of 40m/s. They are fired in the same direction at the same time but separated by 1m. If we say this happens on a co-ordinate system, we can say they are fired from +/- 0.5 x, and parallel to the y axis.

I have worked out that the time of flight is 5.77s and that the y co-ordinates of the impact points are +/- 160.2m.

How can I set up the calculus to solve for x?
 
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The gravitational force is vertical. By symmetry, the electrostatic force is always horizontal, in the x direction. If you're ignoring air resistance, there's no interaction between these. So you can treat the particles as repelling, and moving, purely in the x direction for the known flight time.
Again by symmetry, the repulsion on each ~ 1/x^2:
x'' ~ x^-2
Can you solve that?
 
Yes but neither the acceleration or the separation are constant so I am a bit confused.
 
Quite so. The acceleration (x direction) is as I indicated, is inversely proportional to the square of the separation (x direction). Can you solve that differential equation?
 
Sorry, yes you did show that.
Ok so I solve and get x=-c.ln(x)+x0.
Again I am unsure how to solve this.
 

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