Trajectory function of projectile motion

In summary, the author found the trajectory function by solving the following parametric equation: x(t)=\frac{cp_0cos\theta}{F}ln\left \{ \frac{\sqrt{E_0^2+c^2P^2(t)-c^2p_0^2sen^2\theta}+cP(t)}{E_0-cp_0sen\theta} \right \}\\\\ e-X = ... = (E - cp sinθ)(√Q - cP)/(E2 - c2p2sin2θ)
  • #1
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I've been studying this rather interesting article about projectile motion in special relativity. The thing is, I can't understand how the author found the trajectory function. He says that he did it by solving the following parametric equation:
[tex]x(t)=\frac{cp_0cos\theta}{F}ln\left \{ \frac{\sqrt{E_0^2+c^2P^2(t)-c^2p_0^2sen^2\theta}+cP(t)}{E_0-cp_0sen\theta} \right \}\\\\
y(t)=\frac{1}{F}\left \{ E_0-\sqrt{E_0^2+c^2P^2(t)-c^2p_0^2sen^2\theta} \right \}[/tex]
For which he found the following function:
[tex]y(x)=\frac{E_0}{F}-\frac{E_0}{F}cosh\left [ \frac{Fx}{p_occos\theta} \right ]+\frac{p_0csen\theta}{F}senh\left [ \frac{Fx}{p_0ccos\theta} \right ][/tex]
I'm having some trouble with this calculation because of that [itex]cP(t)[/itex] term. I've tried backtracking as well, but it didn't work. I'm feeling stupid. :(
 
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  • #2
Don't know what any of this stands for, and the paper is behind a paywall. What is F, what is P(t)?? (I assume that sen, senh mean sin, sinh.)

Let X = Fx/(p0c cosθ), let Q = stuff inside the √, and drop the 0 subscripts on p0, E0.

eX = (√Q + cP)/(E - cp sinθ) = (E + cp sinθ)(√Q + cP)/(E2 - c2p2sin2θ)
e-X = ... = (E - cp sinθ)(√Q - cP)/(E2 - c2p2sin2θ)

cosh X = (E√Q + c2pP sin2θ)/(E2 - c2p2sin2θ)

sinh X = (cp sinθ√Q + cPE)/(E2 - c2p2sin2θ)

From which, -E cosh X + pc sinθ sinh X = -√Q, which is (almost) y.
 
  • #3
I'm sorry, I didn't write what any of these terms meant because my issue was with the algebra, not the problem's physics.
[itex]P(t)=Ft-p_0sin\theta[/itex] (I think the author did that to simplify the equations) and I'm assuming [itex]F[/itex] is the weight.

Thank you for the help, it didn't occur to me that if I multiplied [itex]e^{-X}[/itex] by that I would get rid of the [itex]Q[/itex] in the denominator. Like I said, I'm stupid. :(
 

1. What is the trajectory function of projectile motion?

The trajectory function of projectile motion is a mathematical equation that describes the path of an object, such as a ball, that is thrown or launched into the air. It takes into account the initial speed, angle of launch, and the effects of gravity to predict the position of the object at any given time.

2. How is the trajectory function of projectile motion calculated?

The trajectory function of projectile motion is calculated using the equations of motion, which are derived from Newton's laws of motion. These equations take into account the initial velocity, angle of launch, and acceleration due to gravity to determine the position of the object at any given time.

3. What is the importance of understanding the trajectory function of projectile motion?

Understanding the trajectory function of projectile motion is important in many fields, such as physics, engineering, and sports. It allows us to predict the path of an object and make accurate calculations for things like trajectories of rockets, ballistics, and long-range sports like golf and archery.

4. How does air resistance affect the trajectory function of projectile motion?

Air resistance, also known as drag, can affect the trajectory function of projectile motion by slowing down the object's motion and changing its path. This is because air resistance creates a force that acts opposite to the object's motion, causing it to lose speed and change direction. It is often considered negligible for small objects, but it becomes more significant for larger and faster-moving objects.

5. Can the trajectory function of projectile motion be applied to non-linear paths?

No, the trajectory function of projectile motion only applies to linear paths, meaning that the object follows a straight line. Non-linear paths, such as curved or parabolic paths, require different equations and considerations, such as centripetal force and acceleration, to accurately predict the object's motion.

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