Predicting the Trajectory of a Projectile Using Relativity

Relativity, often overshadowed by its simpler cousin, Newtonian physics, offers a more accurate description of the universe when dealing with high velocities or strong gravitational fields. Despite the overwhelming support for classical mechanics in everyday scenarios, relativity provides a crucial framework for predicting the motion of projectiles under relativistic conditions.

When Relativity Merges with Newtonian Physics

The equations of relativity reduce to those of Newtonian physics when the velocity is low and the gravitational field is not too strong. This is why we often use Newton's laws without contradiction; they are accurate in situations where relativistic effects are negligible.

Exploring Relativistic Equations of Motion

There are multiple ways to express the relativistic equations of motion, but one favorite expression is akin to the familiar conservation of energy approach in Newtonian physics:

Elliptical energy balance: KE GPE or 1/2mv^2 mgh

In the context of relativity, the equations take on another form:

Invariance of time: time change due to velocity time change due to gravity
Mathematically, this can be represented as: Delta gamma Delta gamma_g

Where:

gamma 1 - v^2/c^2-0.5 gamma_g 1 - r_s/r-0.5 r_s 2GM/c^2

For instance, the value of r_s for the Earth is approximately 0.00887 meters. The change in elevation from 2000 meters to 1000 meters above the Earth’s radius of 6,370,000 meters gives a Delta gamma_g of 1.09e-13. A stationary object's initial gamma value is 1, and the change to 1.000000000000109 indicates that the final velocity is calculated as:

v c * sqrt{1 – 1/gamma^2} 299,792,458 * sqrt{1 – 1/1.000000000000109^2} 140 m/s

This matches Newton's prediction, confirming that for low velocities, the relativistic and Newtonian approaches yield the same results.

Handling High Velocities and Gravitational Fields

When dealing with relativistic velocities, additional corrections need to be made to account for mass increase and time dilation. For example, in the Large Hadron Collider (LHC), protons moving at high speeds need magnetic fields adjusted to take into account their increased relativistic mass. This adjustment is crucial for maintaining the stability and accuracy of the collider's operations.

For a projectile's trajectory, any initial horizontal or vertical velocity can be handled by the same principles. The key is to consider the relativistic effects on the change in velocity:

If the initial velocity is 300 m/s, the change in gamma comes from 5.01e-13 to 6.10e-13, leading to a final velocity of 331 m/s. If the original velocity is purely horizontal, the vertical component remains 140 m/s, while the horizontal component remains 300 m/s. Trigonometry can be used to determine the resultant velocity for any angle of projection.

For everyday applications, this approach is highly useful. Once solved with a spreadsheet, any Newtonian ballistic problem can be verified, and the classical approach remains accurate until gamma or gamma_g becomes larger than 1.001. At this point, relativity provides a more accurate solution, which is typically applicable only under extreme conditions of high velocity or strong gravity.

Conclusion

While Newtonian physics works perfectly for most everyday scenarios, relativistic physics offers a more comprehensive perspective. Understanding and applying the principles of relativity in predicting projectile trajectories can lead to more precise and accurate models, especially in scenarios involving high speeds or strong gravitational fields.