Discussion Overview
The discussion revolves around solving linear differential equations in the context of kinematics, particularly focusing on equations that describe motion with air resistance. Participants explore methods for integrating these equations, share their approaches, and debate the correct application of mathematical techniques.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant presents the equations \(\ddot{x}(t)+\alpha \dot{x}(t)^2=0\) and \(\ddot{y}(t)+\alpha\dot{y}(t)^2=-g\) and seeks help with the integration involved in solving them.
- Another participant points out that the equations are not linear and suggests rewriting them in terms of velocity.
- Multiple participants express difficulty in solving the integral \(\int \frac{1}{-g-\alpha \dot{y}^2}\:d\dot{y}\) and discuss various substitution methods that lead to complications.
- There is a suggestion to use inverse hyperbolic functions such as arctan and arctanh for integration, with differing opinions on which is appropriate.
- Participants debate the signs of forces acting on the projectile, particularly in relation to gravity and air resistance, and whether they should be considered the same or opposite depending on the direction of motion.
- One participant claims to have solved the ODE using arctan, while another insists that arctanh should have been used instead, leading to further discussion on the correctness of their approaches.
- There is a mention of the importance of separating cases for upward and downward motion, with some participants clarifying their assumptions regarding the forces involved.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the correct method for solving the integral or the appropriate use of arctan versus arctanh. There is also disagreement regarding the treatment of forces acting on the projectile during different phases of motion.
Contextual Notes
Participants express uncertainty about the correct integration techniques and the implications of their assumptions regarding the signs of forces. The discussion reflects a variety of approaches and interpretations without resolving the underlying mathematical challenges.
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You'll get the expression for "arctanh" in terms of "ln"...A useful result...A nice proof to it,also...